Inclusive IVT Witnesses and Mode-Specific Continuity Boundaries
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batch-20261015-advance-20261116-r02-intermediate-value-theorem-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
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Field-order and source-binding data
1. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[12, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-13<b=-11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([12, 1], -13, -11, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([12, 1], -13, -11, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
2. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-12<b=-10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([11, 1], -12, -10, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([11, 1], -12, -10, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
3. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-11<b=-9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([10, 1], -11, -9, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([10, 1], -11, -9, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
4. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-10<b=-8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([9, 1], -10, -8, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([9, 1], -10, -8, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
5. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-9<b=-7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([8, 1], -9, -7, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([8, 1], -9, -7, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
6. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-8<b=-6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([7, 1], -8, -6, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([7, 1], -8, -6, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
7. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-7<b=-5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([6, 1], -7, -5, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([6, 1], -7, -5, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
8. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-6<b=-4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([5, 1], -6, -4, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([5, 1], -6, -4, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
9. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-5<b=-3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([4, 1], -5, -3, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([4, 1], -5, -3, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
10. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-4<b=-2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([3, 1], -4, -2, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([3, 1], -4, -2, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
11. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-3<b=-1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([2, 1], -3, -1, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([2, 1], -3, -1, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
12. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-2<b=0 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([1, 1], -2, 0, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([1, 1], -2, 0, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
13. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[0, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-1<b=1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([0, 1], -1, 1, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([0, 1], -1, 1, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
14. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=0<b=2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-1, 1], 0, 2, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-1, 1], 0, 2, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
15. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=1<b=3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-2, 1], 1, 3, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-2, 1], 1, 3, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
16. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=2<b=4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-3, 1], 2, 4, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-3, 1], 2, 4, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
17. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=3<b=5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-4, 1], 3, 5, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-4, 1], 3, 5, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
18. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=4<b=6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-5, 1], 4, 6, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-5, 1], 4, 6, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
19. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=5<b=7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-6, 1], 5, 7, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-6, 1], 5, 7, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
20. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=6<b=8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-7, 1], 6, 8, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-7, 1], 6, 8, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
21. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=7<b=9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-8, 1], 7, 9, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-8, 1], 7, 9, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
22. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=8<b=10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-9, 1], 8, 10, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-9, 1], 8, 10, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
23. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=9<b=11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-10, 1], 9, 11, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-10, 1], 9, 11, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
24. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=10<b=12 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-11, 1], 10, 12, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-11, 1], 10, 12, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
25. Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations
Coefficients=[-12, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=11<b=13 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).
View answer and explanation
([-12, 1], 11, 13, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)
Evaluate both endpoints exactly and test target-between-values in either orientation, including equality at an endpoint. A target outside the bracket does not satisfy this IVT premise; finite sample agreement does not establish continuity of an arbitrary function. The exact result, retaining the requested order, is ([-12, 1], 11, 13, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-hypothesis · candidate
26. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[12, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-13<b=-11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-12. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-12, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-12, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
27. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-12<b=-10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-11. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-11, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-11, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
28. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-11<b=-9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-10. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-10, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-10, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
29. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-10<b=-8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-9. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-9, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-9, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
30. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-9<b=-7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-8. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-8, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-8, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
31. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-8<b=-6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-7. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-7, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-7, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
32. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-7<b=-5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-6. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-6, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-6, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
33. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-6<b=-4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-5. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-5, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-5, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
34. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-5<b=-3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-4. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-4, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-4, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
35. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-4<b=-2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-3. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-3, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-3, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
36. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-3<b=-1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-2. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-2, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-2, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
37. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-2<b=0 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-1. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(-1, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (-1, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
38. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[0, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-1<b=1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=0. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(0, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (0, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
39. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=0<b=2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=1. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(1, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (1, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
40. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=1<b=3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=2. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(2, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (2, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
41. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=2<b=4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=3. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(3, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (3, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
42. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=3<b=5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=4. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(4, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (4, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
43. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=4<b=6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=5. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(5, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (5, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
44. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=5<b=7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=6. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(6, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (6, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
45. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=6<b=8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=7. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(7, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (7, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
46. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=7<b=9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=8. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(8, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (8, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
47. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=8<b=10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=9. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(9, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (9, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
48. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=9<b=11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=10. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(10, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (10, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
49. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=10<b=12 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=11. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(11, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (11, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
50. An Exact Closed-Interval Root Witness Does Not Assert Uniqueness
Coefficients=[-12, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=11<b=13 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=12. Give (root, value_at_root, inside_closed_interval, hits_target, uniqueness_or_closed_form_claim, certificate).
View answer and explanation
(12, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form)
Check membership in [a,b] and f(root)=target exactly, along with the bracket. The fixed No withholds uniqueness or a rational closed-form guarantee in the general theorem; it does not negate this supplied rational witness. The exact result, retaining the requested order, is (12, 0, Yes, Yes, No, IVT_existence_does_not_imply_uniqueness_or_rational_closed_form).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-exact-root · candidate
51. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[13, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-13<b=-11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-13. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-13, -11, 0, -13, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-13, -11, 0, -13, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
52. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[12, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-12<b=-10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-12. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-12, -10, 0, -12, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-12, -10, 0, -12, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
53. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-11<b=-9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-11. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-11, -9, 0, -11, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-11, -9, 0, -11, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
54. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-10<b=-8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-10. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-10, -8, 0, -10, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-10, -8, 0, -10, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
55. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-9<b=-7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-9. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-9, -7, 0, -9, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-9, -7, 0, -9, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
56. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-8<b=-6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-8. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-8, -6, 0, -8, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-8, -6, 0, -8, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
57. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-7<b=-5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-7. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-7, -5, 0, -7, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-7, -5, 0, -7, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
58. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-6<b=-4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-6. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-6, -4, 0, -6, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-6, -4, 0, -6, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
59. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-5<b=-3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-5. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-5, -3, 0, -5, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-5, -3, 0, -5, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
60. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-4<b=-2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-4. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-4, -2, 0, -4, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-4, -2, 0, -4, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
61. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-3<b=-1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-3. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-3, -1, 0, -3, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-3, -1, 0, -3, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
62. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-2<b=0 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-2. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-2, 0, 0, -2, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-2, 0, 0, -2, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
63. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-1<b=1 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=-1. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(-1, 1, 0, -1, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (-1, 1, 0, -1, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
64. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[0, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=0<b=2 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=0. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(0, 2, 0, 0, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (0, 2, 0, 0, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
65. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-1, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=1<b=3 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=1. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(1, 3, 0, 1, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (1, 3, 0, 1, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
66. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-2, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=2<b=4 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=2. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(2, 4, 0, 2, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (2, 4, 0, 2, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
67. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-3, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=3<b=5 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=3. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(3, 5, 0, 3, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (3, 5, 0, 3, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
68. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-4, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=4<b=6 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=4. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(4, 6, 0, 4, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (4, 6, 0, 4, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
69. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-5, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=5<b=7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=5. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(5, 7, 0, 5, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (5, 7, 0, 5, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
70. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-6, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=6<b=8 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=6. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(6, 8, 0, 6, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (6, 8, 0, 6, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
71. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-7, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=7<b=9 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=7. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(7, 9, 0, 7, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (7, 9, 0, 7, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
72. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=8<b=10 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=8. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(8, 10, 0, 8, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (8, 10, 0, 8, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
73. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-9, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=9<b=11 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=9. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(9, 11, 0, 9, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (9, 11, 0, 9, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
74. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-10, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=10<b=12 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=10. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(10, 12, 0, 10, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (10, 12, 0, 10, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
75. The Inclusive Bracket Allows an Endpoint Root
Coefficients=[-11, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=11<b=13 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Supplied exact root=11. Give (a, b, target, root, root_is_endpoint, inclusive_bracket, root_certificate, certificate).
View answer and explanation
(11, 13, 0, 11, Yes, Yes, Yes, endpoints_are_included)
Retain endpoint/root equality as a separate flag. The closed interval includes either endpoint, but an interior root can also satisfy the same family. An exact witness is not a claim that finite bisection has converged. The exact result, retaining the requested order, is (11, 13, 0, 11, Yes, Yes, Yes, endpoints_are_included).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-endpoint-bracket · candidate
76. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=discontinuous_jump and positive exact scale=1. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(discontinuous_jump, f(x)=-1_for_x<0_and_1_for_x>=0, -1, 1, 0, Yes, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-1_for_x<0_and_1_for_x>=0, -1, 1, 0, Yes, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
77. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonbracket and positive exact scale=2. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonbracket, f(x)=2*x, 0, 1, 4, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=2*x, 0, 1, 4, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
78. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonunique and positive exact scale=3. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonunique, f(x)=x^2, -3, 3, 9, Yes, two_roots_possible, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -3, 3, 9, Yes, two_roots_possible, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
79. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=discontinuous_jump and positive exact scale=4. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(discontinuous_jump, f(x)=-4_for_x<0_and_4_for_x>=0, -1, 1, 0, Yes, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-4_for_x<0_and_4_for_x>=0, -1, 1, 0, Yes, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
80. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonbracket and positive exact scale=5. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonbracket, f(x)=5*x, 0, 1, 10, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=5*x, 0, 1, 10, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
81. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonunique and positive exact scale=6. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonunique, f(x)=x^2, -6, 6, 36, Yes, two_roots_possible, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -6, 6, 36, Yes, two_roots_possible, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
82. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=discontinuous_jump and positive exact scale=7. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(discontinuous_jump, f(x)=-7_for_x<0_and_7_for_x>=0, -1, 1, 0, Yes, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-7_for_x<0_and_7_for_x>=0, -1, 1, 0, Yes, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
83. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonbracket and positive exact scale=8. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonbracket, f(x)=8*x, 0, 1, 16, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=8*x, 0, 1, 16, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
84. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonunique and positive exact scale=9. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonunique, f(x)=x^2, -9, 9, 81, Yes, two_roots_possible, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -9, 9, 81, Yes, two_roots_possible, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
85. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=discontinuous_jump and positive exact scale=10. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(discontinuous_jump, f(x)=-10_for_x<0_and_10_for_x>=0, -1, 1, 0, Yes, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-10_for_x<0_and_10_for_x>=0, -1, 1, 0, Yes, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
86. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonbracket and positive exact scale=11. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonbracket, f(x)=11*x, 0, 1, 22, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=11*x, 0, 1, 22, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
87. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonunique and positive exact scale=12. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonunique, f(x)=x^2, -12, 12, 144, Yes, two_roots_possible, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -12, 12, 144, Yes, two_roots_possible, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
88. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=discontinuous_jump and positive exact scale=13. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(discontinuous_jump, f(x)=-13_for_x<0_and_13_for_x>=0, -1, 1, 0, Yes, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-13_for_x<0_and_13_for_x>=0, -1, 1, 0, Yes, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
89. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonbracket and positive exact scale=14. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonbracket, f(x)=14*x, 0, 1, 28, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=14*x, 0, 1, 28, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
90. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonunique and positive exact scale=15. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonunique, f(x)=x^2, -15, 15, 225, Yes, two_roots_possible, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -15, 15, 225, Yes, two_roots_possible, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
91. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=discontinuous_jump and positive exact scale=16. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(discontinuous_jump, f(x)=-16_for_x<0_and_16_for_x>=0, -1, 1, 0, Yes, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-16_for_x<0_and_16_for_x>=0, -1, 1, 0, Yes, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
92. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonbracket and positive exact scale=17. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonbracket, f(x)=17*x, 0, 1, 34, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=17*x, 0, 1, 34, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
93. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonunique and positive exact scale=18. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonunique, f(x)=x^2, -18, 18, 324, Yes, two_roots_possible, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -18, 18, 324, Yes, two_roots_possible, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
94. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=discontinuous_jump and positive exact scale=19. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(discontinuous_jump, f(x)=-19_for_x<0_and_19_for_x>=0, -1, 1, 0, Yes, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-19_for_x<0_and_19_for_x>=0, -1, 1, 0, Yes, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
95. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonbracket and positive exact scale=20. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonbracket, f(x)=20*x, 0, 1, 40, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=20*x, 0, 1, 40, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
96. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonunique and positive exact scale=21. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonunique, f(x)=x^2, -21, 21, 441, Yes, two_roots_possible, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -21, 21, 441, Yes, two_roots_possible, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
97. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=discontinuous_jump and positive exact scale=22. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(discontinuous_jump, f(x)=-22_for_x<0_and_22_for_x>=0, -1, 1, 0, Yes, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-22_for_x<0_and_22_for_x>=0, -1, 1, 0, Yes, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
98. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonbracket and positive exact scale=23. Give (mode, function_description, a, b, target, inclusive_bracket, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonbracket, f(x)=23*x, 0, 1, 46, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonbracket, f(x)=23*x, 0, 1, 46, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
99. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=nonunique and positive exact scale=24. Give (mode, function_description, a, b, target, inclusive_bracket, multiplicity_statement, uniqueness_claim).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(nonunique, f(x)=x^2, -24, 24, 576, Yes, two_roots_possible, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (nonunique, f(x)=x^2, -24, 24, 576, Yes, two_roots_possible, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
100. A Discontinuous Sign Jump and Failure of Uniqueness Have Different Boundaries
Use boundary mode=discontinuous_jump and positive exact scale=25. Give (mode, function_description, a, b, target, inclusive_bracket, continuous, root_exists).
The three modes have8/7/8 fields: jump has no root, nonbracket fails the bracket, and nonunique has two endpoint roots. The final No has different roles; arity cannot choose the branch.
View answer and explanation
(discontinuous_jump, f(x)=-25_for_x<0_and_25_for_x>=0, -1, 1, 0, Yes, No, No)
The discontinuous jump takes only ±scale and misses0 despite bracketing it on [−1,1]. A scaled identity on [0,1] misses target2*scale, so the bracket fails. For x^2 on [−scale,scale], target scale^2 has both endpoint roots: existence is not uniqueness. Preserve each branch field meaning and do not infer continuity from samples. The exact result, retaining the requested order, is (discontinuous_jump, f(x)=-25_for_x<0_and_25_for_x>=0, -1, 1, 0, Yes, No, No).
Reviewed step-by-step derivation
Intermediate value theorem · qf-mod47-ivt-discontinuity-boundary · candidate
Canonical question JSONL · English JSONL · Validation report