# Question 11853: One Exact Jensen Instance: a Counterexample or Only an Example [Return to catalogue page 60](https://ocme.evemisslab.com/en/questions/page/60/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261025-r02-convex-function-foundations/#q-20261013-advance-20261025-r02-convex-function-0078) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0022-11853 · q-20261013-advance-20261025-r02-convex-function-0078 ## Original question and givens Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=-1/1, lambda=3/5 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it. - G001 — Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, -10, -1], not ascending-power order. The declared interval [lower,upper]=[-5, 5] belongs to the function data. x=0/1, y=-1/1, lambda=3/5 must lie in the declared domain and closed unit weight interval. Give (coefficients,interval,x,y,lambda,weighted input,lhs=f(weighted input),rhs=weighted endpoint values,rhs−lhs,inequality passes?,nonconvexity_witness if the gap is negative, otherwise single_instance_only). Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. A positive convexity certificate must cover all endpoint pairs in the convex domain and every lambda in [0,1]. One Jensen violation can refute convexity, but one successful instance cannot prove it. ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-convex-function](https://ocme.evemisslab.com/en/mko/mko-convex-function/) ## Step-by-step derivation S001 · impliesBefore declared interval record=[-5, 5] After ordered_nonempty=Yes;nondegenerate=Yes Conditions: none This nonempty-interval protocol uses lower≤upper;the nondegenerate certificate additionally needs lower=0;5-z=lambda*(5-x)+(1-lambda)*(5-y)>=0 Conditions: none Both slacks are nonnegative sums,hence lower≤z≤upper for every endpoint pair and every permitted lambda.No sampling is used;endpoints and singleton cases are included. Step identifiers and dependenciesRule: interval_all_segments_slack_identity · Uses: G001, S001 S003 · impliesBefore univariate quadratic coefficients=[2, -10, -1];order=[a,b,c] After f(t)=(2)*t^2+(-10)*t+(-1) Conditions: none These are quadratic,linear,constant coefficients in descending order,not the constant-first boundary-polynomial order.Retain zero leading coefficients. Step identifiers and dependenciesRule: quadratic_descending_coefficient_order · Uses: G001, S002 S004 · impliesBefore f(t)=(2)*t^2+(-10)*t+(-1);z=lambda*x+(1-lambda)*y After rhs-lhs=(2)*(lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2) Conditions: none Expand both Jensen sides.The unit weight sum cancels the linear and constant terms exactly,leaving only the quadratic part. Step identifiers and dependenciesRule: jensen_linear_and_constant_terms_cancel · Uses: G001, S003 S005 · impliesBefore lambda*x^2+(1-lambda)*y^2-(lambda*x+(1-lambda)*y)^2 After lambda*(1-lambda)*(x-y)^2;full gap=(2)*lambda*(1-lambda)*(x-y)^2 Conditions: none Expand the square and collect x²,xy,y².The coefficients are lambda(1−lambda),−2lambda(1−lambda),lambda(1−lambda),yielding the(x−y)^2 factor for all real endpoints. Step identifiers and dependenciesRule: jensen_gap_full_symbolic_factorization · Uses: G001, S004 S006 · calculationBefore Jensen complementary weight := sub(1, 3/5) After Jensen complementary weight = 2/5 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Jensen weighted x := mul(3/5, 0) After Jensen weighted x = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Jensen weighted y := mul(2/5, -1) After Jensen weighted y = -2/5 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Jensen weighted input := sum(0, -2/5) After Jensen weighted input = -2/5 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Jensen lhs point squared := mul(-2/5, -2/5) After Jensen lhs point squared = 4/25 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Jensen lhs quadratic term := mul(2, 4/25) After Jensen lhs quadratic term = 8/25 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Jensen lhs linear term := mul(-10, -2/5) After Jensen lhs linear term = 4 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Jensen lhs polynomial value := sum(8/25, 4, -1) After Jensen lhs polynomial value = 83/25 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Jensen left endpoint point squared := mul(0, 0) After Jensen left endpoint point squared = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Jensen left endpoint quadratic term := mul(2, 0) After Jensen left endpoint quadratic term = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Jensen left endpoint linear term := mul(-10, 0) After Jensen left endpoint linear term = 0 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Jensen left endpoint polynomial value := sum(0, 0, -1) After Jensen left endpoint polynomial value = -1 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Jensen right endpoint point squared := mul(-1, -1) After Jensen right endpoint point squared = 1 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Jensen right endpoint quadratic term := mul(2, 1) After Jensen right endpoint quadratic term = 2 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Jensen right endpoint linear term := mul(-10, -1) After Jensen right endpoint linear term = 10 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Jensen right endpoint polynomial value := sum(2, 10, -1) After Jensen right endpoint polynomial value = 11 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Jensen weighted left value := mul(3/5, -1) After Jensen weighted left value = -3/5 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Jensen weighted right value := mul(2/5, 11) After Jensen weighted right value = 22/5 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Jensen rhs := sum(-3/5, 22/5) After Jensen rhs = 19/5 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Jensen rhs minus lhs := sub(19/5, 83/25) After Jensen rhs minus lhs = 12/25 Conditions: none Compute exactly from this step's original roles and established intermediates,retaining signs,denominators and coefficient order. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore qualified x=0;y=-1;lambda=3/5;lhs=83/25;rhs=19/5;gap=12/25 After Jensen instance passes=Yes;scope=single_instance_only Conditions: none A negative gap at qualified endpoints and weight is a counterexample.Nonnegative proves only this instance;lambda0 or x=y can pass even for a nonconvex quadratic. Step identifiers and dependenciesRule: one_Jensen_instance_not_universal_positive_proof · Uses: G001, S025 ## Reverse checks - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Recompute Jensen rhs minus lhs with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Jensen rhs with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Jensen weighted right value with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Jensen weighted left value with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Jensen right endpoint polynomial value with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Jensen right endpoint linear term with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Jensen right endpoint quadratic term with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Jensen right endpoint point squared with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Jensen left endpoint polynomial value with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Jensen left endpoint linear term with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Jensen left endpoint quadratic term with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Jensen left endpoint point squared with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Jensen lhs polynomial value with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Jensen lhs linear term with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Jensen lhs quadratic term with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Jensen lhs point squared with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Jensen weighted input with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Jensen weighted y with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Jensen weighted x with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Jensen complementary weight with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([2, -10, -1], [-5, 5], 0, -1, 3/5, -2/5, 83/25, 19/5, 12/25, Yes, single_instance_only) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 074287d8886ad35a4fe2923b1c1bb220a6f3e9d20341800d21bb0c3806f29cbe - candidate record SHA-256: fee32b518690a1e9bb9b089ed362477de4dd0062a18a755e3a39c3a30692bb44 - semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac - question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261025-r02-convex-function-0078.json)