Universal Affine/Quadratic Convexity Certificates versus Jensen Instances
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261025-r02-convex-function-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-2, 3]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-2, 3], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, 3], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
2. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-3, 4]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-3, 4], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 4], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
3. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-4, 5]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-4, 5], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 5], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
4. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-5, 6]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-5, 6], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 6], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
5. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-6, 7]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-6, 7], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, 7], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
6. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-7, 8]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-7, 8], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, 8], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
7. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-8, 9]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-8, 9], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 9], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
8. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-9, 10]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-9, 10], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 10], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
9. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-10, 11]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-10, 11], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 11], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
10. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-11, 12]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-11, 12], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, 12], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
11. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-12, 13]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-12, 13], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, 13], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
12. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-13, 14]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-13, 14], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-13, 14], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
13. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-14, 15]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-14, 15], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-14, 15], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
14. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-15, 16]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-15, 16], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-15, 16], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
15. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-16, 17]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-16, 17], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-16, 17], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
16. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-17, 18]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-17, 18], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-17, 18], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
17. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-18, 19]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-18, 19], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-18, 19], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
18. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-19, 20]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-19, 20], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-19, 20], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
19. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-20, 21]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-20, 21], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-20, 21], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
20. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-21, 22]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-21, 22], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-21, 22], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
21. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-22, 23]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-22, 23], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-22, 23], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
22. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-23, 24]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-23, 24], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-23, 24], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
23. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-24, 25]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-24, 25], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-24, 25], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
24. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-25, 26]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-25, 26], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-25, 26], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
25. Qualify a Declared Nonempty Interval before Using Jensen Semantics
The interval record is [lower,upper]=[-26, 27]. Give (interval,lower≤upper?,qualified convex interval?,all segment points remain in the interval). A reversed record is unqualified for this nonempty-interval protocol; its No flag does not classify the abstract empty set as nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-26, 27], Yes, Yes, all segment points remain in the interval)
For an ordered interval, arbitrary endpoint pairs and weights in [0,1] stay between its bounds. A singleton is allowed here. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-26, 27], Yes, Yes, all segment points remain in the interval).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-convex-domain-qualification · candidate
26. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-12, -2] and interval=[-2, 3]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-12, -2], [-2, 3], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-12, -2], [-2, 3], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
27. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-11, -1] and interval=[-3, 4]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-11, -1], [-3, 4], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-11, -1], [-3, 4], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
28. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-10, 0] and interval=[-4, 5]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-10, 0], [-4, 5], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-10, 0], [-4, 5], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
29. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-9, 1] and interval=[-5, 6]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-9, 1], [-5, 6], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-9, 1], [-5, 6], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
30. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-8, 2] and interval=[-6, 7]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-8, 2], [-6, 7], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-8, 2], [-6, 7], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
31. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-7, -2] and interval=[-7, 8]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-7, -2], [-7, 8], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-7, -2], [-7, 8], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
32. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-6, -1] and interval=[-8, 9]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-6, -1], [-8, 9], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-6, -1], [-8, 9], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
33. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-5, 0] and interval=[-9, 10]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-5, 0], [-9, 10], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-5, 0], [-9, 10], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
34. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-4, 1] and interval=[-10, 11]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-4, 1], [-10, 11], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-4, 1], [-10, 11], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
35. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-3, 2] and interval=[-11, 12]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-3, 2], [-11, 12], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-3, 2], [-11, 12], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
36. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-2, -2] and interval=[-12, 13]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-2, -2], [-12, 13], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-2, -2], [-12, 13], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
37. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[-1, -1] and interval=[-13, 14]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, -1], [-13, 14], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -1], [-13, 14], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
38. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[0, 0] and interval=[-14, 15]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([0, 0], [-14, 15], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0], [-14, 15], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
39. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[1, 1] and interval=[-15, 16]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, 1], [-15, 16], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1], [-15, 16], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
40. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[2, 2] and interval=[-16, 17]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([2, 2], [-16, 17], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2], [-16, 17], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
41. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[3, -2] and interval=[-17, 18]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, -2], [-17, 18], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -2], [-17, 18], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
42. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[4, -1] and interval=[-18, 19]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([4, -1], [-18, 19], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([4, -1], [-18, 19], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
43. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[5, 0] and interval=[-19, 20]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([5, 0], [-19, 20], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([5, 0], [-19, 20], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
44. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[6, 1] and interval=[-20, 21]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([6, 1], [-20, 21], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([6, 1], [-20, 21], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
45. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[7, 2] and interval=[-21, 22]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([7, 2], [-21, 22], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([7, 2], [-21, 22], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
46. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[8, -2] and interval=[-22, 23]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([8, -2], [-22, 23], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([8, -2], [-22, 23], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
47. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[9, -1] and interval=[-23, 24]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([9, -1], [-23, 24], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([9, -1], [-23, 24], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
48. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[10, 0] and interval=[-24, 25]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([10, 0], [-24, 25], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([10, 0], [-24, 25], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
49. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[11, 1] and interval=[-25, 26]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([11, 1], [-25, 26], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([11, 1], [-25, 26], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
50. An Affine Jensen Identity on a Qualified Interval
Use f(t)=a*t+b with coefficients [a,b]=[12, 2] and interval=[-26, 27]. Give (coefficients,interval,ordered nonempty domain?,f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y),qualified convex certificate?). A singleton is allowed; a reversed interval record is unqualified. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([12, 2], [-26, 27], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes)
Distributivity proves the equality for every pair of domain points and every permitted weight. No finite sampling is used. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([12, 2], [-26, 27], Yes, f(lambda*x+(1-lambda)*y)=lambda*f(x)+(1-lambda)*f(y), Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-affine-universal-certificate · candidate
51. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -12, -3], not ascending-power order. The declared interval [lower,upper]=[-2, 3] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, -12, -3], [-2, 3], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -12, -3], [-2, 3], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
52. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -11, -2], not ascending-power order. The declared interval [lower,upper]=[-3, 4] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, -11, -2], [-3, 4], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -11, -2], [-3, 4], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
53. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-4, 5] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([2, -10, -1], [-4, 5], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -10, -1], [-4, 5], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
54. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -9, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 6] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, -9, 0], [-5, 6], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -9, 0], [-5, 6], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
55. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -8, 1], not ascending-power order. The declared interval [lower,upper]=[-6, 7] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([0, -8, 1], [-6, 7], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -8, 1], [-6, 7], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
56. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -7, 2], not ascending-power order. The declared interval [lower,upper]=[-7, 8] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, -7, 2], [-7, 8], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -7, 2], [-7, 8], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
57. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -6, 3], not ascending-power order. The declared interval [lower,upper]=[-8, 9] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([2, -6, 3], [-8, 9], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -6, 3], [-8, 9], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
58. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -5, -3], not ascending-power order. The declared interval [lower,upper]=[-9, 10] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, -5, -3], [-9, 10], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -5, -3], [-9, 10], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
59. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -4, -2], not ascending-power order. The declared interval [lower,upper]=[-10, 11] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([0, -4, -2], [-10, 11], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -4, -2], [-10, 11], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
60. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -3, -1], not ascending-power order. The declared interval [lower,upper]=[-11, 12] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, -3, -1], [-11, 12], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -3, -1], [-11, 12], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
61. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -2, 0], not ascending-power order. The declared interval [lower,upper]=[-12, 13] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, -2, 0], [-12, 13], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -2, 0], [-12, 13], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
62. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -1, 1], not ascending-power order. The declared interval [lower,upper]=[-13, 14] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, -1, 1], [-13, 14], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -1, 1], [-13, 14], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
63. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 0, 2], not ascending-power order. The declared interval [lower,upper]=[-14, 15] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([0, 0, 2], [-14, 15], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 0, 2], [-14, 15], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
64. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 1, 3], not ascending-power order. The declared interval [lower,upper]=[-15, 16] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, 1, 3], [-15, 16], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3], [-15, 16], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
65. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 2, -3], not ascending-power order. The declared interval [lower,upper]=[-16, 17] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([2, 2, -3], [-16, 17], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3], [-16, 17], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
66. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 3, -2], not ascending-power order. The declared interval [lower,upper]=[-17, 18] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, 3, -2], [-17, 18], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 3, -2], [-17, 18], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
67. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 4, -1], not ascending-power order. The declared interval [lower,upper]=[-18, 19] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([0, 4, -1], [-18, 19], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 4, -1], [-18, 19], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
68. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 5, 0], not ascending-power order. The declared interval [lower,upper]=[-19, 20] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, 5, 0], [-19, 20], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 5, 0], [-19, 20], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
69. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 6, 1], not ascending-power order. The declared interval [lower,upper]=[-20, 21] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([2, 6, 1], [-20, 21], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 6, 1], [-20, 21], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
70. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 7, 2], not ascending-power order. The declared interval [lower,upper]=[-21, 22] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, 7, 2], [-21, 22], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 7, 2], [-21, 22], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
71. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 8, 3], not ascending-power order. The declared interval [lower,upper]=[-22, 23] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, 8, 3], [-22, 23], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 8, 3], [-22, 23], Yes, -1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, No).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
72. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 9, -3], not ascending-power order. The declared interval [lower,upper]=[-23, 24] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, 9, -3], [-23, 24], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 9, -3], [-23, 24], Yes, 1, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
73. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-24, 25] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([2, 10, -2], [-24, 25], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 10, -2], [-24, 25], Yes, 2, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
74. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 11, -1], not ascending-power order. The declared interval [lower,upper]=[-25, 26] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, 11, -1], [-25, 26], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 11, -1], [-25, 26], Yes, 3, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
75. A Nondegenerate-Interval Quadratic Jensen-Gap Certificate
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 12, 0], not ascending-power order. The declared interval [lower,upper]=[-26, 27] belongs to the function data. Give (coefficients,interval,lower<upper?,leading coefficient,rhs-lhs=a*lambda*(1-lambda)*(x-y)^2,qualified convex certificate?). This family requires a nondegenerate interval and a≥0. A No for a singleton or reversed interval means this certificate is unqualified, not that the function on a singleton is nonconvex. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([0, 12, 0], [-26, 27], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes)
Expand the Jensen gap symbolically. On a nondegenerate interval its sign is controlled by a, since lambda*(1−lambda)*(x−y)² is nonnegative. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 12, 0], [-26, 27], Yes, 0, rhs-lhs=a*lambda*(1-lambda)*(x-y)^2, Yes).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-quadratic-universal-certificate · candidate
76. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -12, -3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=-3/1, lambda=1/3 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, -12, -3], [-5, 5], -2, -3, 1/3, -8/3, 197/9, 65/3, -2/9, No, nonconvexity_witness)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -12, -3], [-5, 5], -2, -3, 1/3, -8/3, 197/9, 65/3, -2/9, No, nonconvexity_witness).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
77. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -11, -2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=-2/1, lambda=2/4 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, -11, -2], [-5, 5], -1, -2, 1/2, -3/2, 67/4, 17, 1/4, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -11, -2], [-5, 5], -1, -2, 1/2, -3/2, 67/4, 17, 1/4, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
78. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=-1/1, lambda=3/5 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([2, -10, -1], [-5, 5], 0, -1, 3/5, -2/5, 83/25, 19/5, 12/25, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -10, -1], [-5, 5], 0, -1, 3/5, -2/5, 83/25, 19/5, 12/25, Yes, single_instance_only).
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Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
79. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -9, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=1/1, y=0/1, lambda=4/6 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, -9, 0], [-5, 5], 1, 0, 2/3, 2/3, -14/3, -4, 2/3, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -9, 0], [-5, 5], 1, 0, 2/3, 2/3, -14/3, -4, 2/3, Yes, single_instance_only).
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Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
80. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -8, 1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=2/1, y=1/1, lambda=5/7 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([0, -8, 1], [-5, 5], 2, 1, 5/7, 12/7, -89/7, -89/7, 0, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -8, 1], [-5, 5], 2, 1, 5/7, 12/7, -89/7, -89/7, 0, Yes, single_instance_only).
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Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
81. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -7, 2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=2/1, lambda=6/8 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, -7, 2], [-5, 5], -2, 2, 3/4, -1, 10, 13, 3, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -7, 2], [-5, 5], -2, 2, 3/4, -1, 10, 13, 3, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
82. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, -6, 3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=3/1, lambda=7/9 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, No, nonconvexity_witness)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, -6, 3], [-5, 5], -1, 3, 7/9, -1/9, 296/81, 8/9, -224/81, No, nonconvexity_witness).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
83. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -5, -3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=-3/1, lambda=8/10 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, -5, -3], [-5, 5], 0, -3, 4/5, -3/5, 27/25, 27/5, 108/25, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -5, -3], [-5, 5], 0, -3, 4/5, -3/5, 27/25, 27/5, 108/25, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
84. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, -4, -2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=1/1, y=-2/1, lambda=9/11 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([0, -4, -2], [-5, 5], 1, -2, 9/11, 5/11, -42/11, -42/11, 0, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, -4, -2], [-5, 5], 1, -2, 9/11, 5/11, -42/11, -42/11, 0, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
85. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, -3, -1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=2/1, y=-1/1, lambda=10/12 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, -3, -1], [-5, 5], 2, -1, 5/6, 3/2, -13/4, -2, 5/4, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, -3, -1], [-5, 5], 2, -1, 5/6, 3/2, -13/4, -2, 5/4, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
86. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -2, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=0/1, lambda=11/13 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([2, -2, 0], [-5, 5], -2, 0, 11/13, -22/13, 1540/169, 132/13, 176/169, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, -2, 0], [-5, 5], -2, 0, 11/13, -22/13, 1540/169, 132/13, 176/169, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
87. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, -1, 1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=1/1, lambda=12/14 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, -1, 1], [-5, 5], -1, 1, 6/7, -5/7, 159/49, 33/7, 72/49, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, -1, 1], [-5, 5], -1, 1, 6/7, -5/7, 159/49, 33/7, 72/49, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
88. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 0, 2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=2/1, lambda=13/15 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, 0, 2], [-5, 5], 0, 2, 13/15, 4/15, 434/225, 22/15, -104/225, No, nonconvexity_witness)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 0, 2], [-5, 5], 0, 2, 13/15, 4/15, 434/225, 22/15, -104/225, No, nonconvexity_witness).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
89. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 1, 3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=1/1, y=3/1, lambda=14/16 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, 1, 3], [-5, 5], 1, 3, 7/8, 5/4, 93/16, 25/4, 7/16, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 1, 3], [-5, 5], 1, 3, 7/8, 5/4, 93/16, 25/4, 7/16, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
90. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 2, -3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=2/1, y=-3/1, lambda=15/17 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([2, 2, -3], [-5, 5], 2, -3, 15/17, 24/17, 1101/289, 9, 1500/289, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 2, -3], [-5, 5], 2, -3, 15/17, 24/17, 1101/289, 9, 1500/289, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
91. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 3, -2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=-2/1, lambda=16/18 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, 3, -2], [-5, 5], -2, -2, 8/9, -2, 4, 4, 0, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 3, -2], [-5, 5], -2, -2, 8/9, -2, 4, 4, 0, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
92. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 4, -1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=-1/1, lambda=17/19 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([0, 4, -1], [-5, 5], -1, -1, 17/19, -1, -5, -5, 0, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 4, -1], [-5, 5], -1, -1, 17/19, -1, -5, -5, 0, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
93. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 5, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=0/1, lambda=18/20 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, 5, 0], [-5, 5], 0, 0, 9/10, 0, 0, 0, 0, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 5, 0], [-5, 5], 0, 0, 9/10, 0, 0, 0, 0, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
94. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 6, 1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=1/1, y=1/1, lambda=19/21 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, 6, 1], [-5, 5], 1, 1, 19/21, 1, 6, 6, 0, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 6, 1], [-5, 5], 1, 1, 19/21, 1, 6, 6, 0, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
95. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 7, 2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=2/1, y=2/1, lambda=20/22 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, 7, 2], [-5, 5], 2, 2, 10/11, 2, 28, 28, 0, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 7, 2], [-5, 5], 2, 2, 10/11, 2, 28, 28, 0, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
96. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[0, 8, 3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-2/1, y=3/1, lambda=21/23 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([0, 8, 3], [-5, 5], -2, 3, 21/23, -36/23, -219/23, -219/23, 0, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([0, 8, 3], [-5, 5], -2, 3, 21/23, -36/23, -219/23, -219/23, 0, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
97. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[1, 9, -3], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=-1/1, y=-3/1, lambda=22/24 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([1, 9, -3], [-5, 5], -1, -3, 11/12, -7/6, -437/36, -71/6, 11/36, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([1, 9, -3], [-5, 5], -1, -3, 11/12, -7/6, -437/36, -71/6, 11/36, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
98. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=-2/1, lambda=23/25 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([2, 10, -2], [-5, 5], 0, -2, 23/25, -4/25, -2218/625, -74/25, 368/625, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([2, 10, -2], [-5, 5], 0, -2, 23/25, -4/25, -2218/625, -74/25, 368/625, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
99. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[3, 11, -1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=1/1, y=-1/1, lambda=24/26 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([3, 11, -1], [-5, 5], 1, -1, 12/13, 11/13, 1767/169, 147/13, 144/169, Yes, single_instance_only)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([3, 11, -1], [-5, 5], 1, -1, 12/13, 11/13, 1767/169, 147/13, 144/169, Yes, single_instance_only).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
100. One Exact Jensen Instance: a Counterexample or Only an Example
Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[-1, 12, 0], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=2/1, y=0/1, lambda=25/27 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it.
View answer and explanation
([-1, 12, 0], [-5, 5], 2, 0, 25/27, 50/27, 13700/729, 500/27, -200/729, No, nonconvexity_witness)
Compute both sides exactly. A negative gap refutes convexity on the declared domain; a nonnegative gap proves only this one instance. This exact finite slice does not automatically establish a general infinite conclusion. The exact result, retaining the requested order, is ([-1, 12, 0], [-5, 5], 2, 0, 25/27, 50/27, 13700/729, 500/27, -200/729, No, nonconvexity_witness).
Reviewed step-by-step derivation
Convex function · qf-mod25-convexfn-jensen-instance-audit · candidate
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