# Question 11948: Evaluate the Polynomial while Separately Reporting Feasibility

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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.

R0022-11948 · q-20261013-advance-20261025-r03-convex-optimization-problem-0073



## Original question and givens

Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-24, 25] belongs to the function data. point=-2/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.


- G001 — Use the one-variable quadratic f(t)=a*t²+b*t+c with coefficients [a,b,c]=[2, 10, -2], not ascending-power order. The declared interval [lower,upper]=[-24, 25] belongs to the function data. point=-2/1. Give (coefficients,interval,normalized point,feasible?,polynomial value,evaluation_only). Retain the polynomial value even for an infeasible candidate; its numerical value does not make it eligible. Use exact rational arithmetic with reduced positive denominators; omit /1 for integers. Keep convex-objective qualification, convex feasible-domain qualification, point feasibility, evaluation and comparison separate. These checks supply no general solver, global optimum, existence, uniqueness, KKT or convergence claim.





## 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-optimization-problem](https://ocme.evemisslab.com/en/mko/mko-convex-optimization-problem/)





## Step-by-step derivation

S001 · impliesBefore

declared interval record=[-24, 25]

After

ordered_nonempty=Yes;nondegenerate=Yes

Conditions: none

This nonempty-interval protocol uses lower≤upper;the nondegenerate certificate additionally needs lower<upper.A reversed record is unqualified,not a declaration that the abstract empty set is nonconvex.A singleton interval is convex.

Step identifiers and dependenciesRule: interval_certificate_qualification_not_empty_set_nonconvexity · Uses: G001





S002 · impliesBefore

univariate quadratic coefficients=[2, 10, -2];order=[a,b,c]

After

f(t)=(2)*t^2+(10)*t+(-2)

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, S001





S003 · calculationBefore

candidate lower slack := sub(-2, -24)

After

candidate lower slack = 22

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, S002





S004 · calculationBefore

candidate upper slack := sub(25, -2)

After

candidate upper slack = 27

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, S003





S005 · impliesBefore

candidate:interval=[-24, 25];point=-2;slacks=[22, 27]

After

feasible=Yes

Conditions: none

Both inclusive bounds must hold.A reversed record supplies no qualified feasible point.The polynomial may have a value outside the domain,but a value does not establish eligibility.

Step identifiers and dependenciesRule: inclusive_point_feasibility_separate_evaluation · Uses: G001, S004





S006 · calculationBefore

candidate objective point squared := mul(-2, -2)

After

candidate objective point squared = 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, S005





S007 · calculationBefore

candidate objective quadratic term := mul(2, 4)

After

candidate objective quadratic term = 8

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

candidate objective linear term := mul(10, -2)

After

candidate objective linear term = -20

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

candidate objective polynomial value := sum(8, -20, -2)

After

candidate objective polynomial value = -14

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 · impliesBefore

point=-2;feasible=Yes;polynomial value=-14

After

retain value even for infeasible point;evaluation is not optimization

Conditions: none

Retain the polynomial value even when the point is infeasible,as specified.An outside value does not become a feasible solution or optimum.

Step identifiers and dependenciesRule: outside_domain_value_not_feasibility_or_optimality · Uses: G001, S009









## Reverse checks


- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Recompute candidate objective polynomial value with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute candidate objective linear term with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute candidate objective quadratic term with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute candidate objective point squared 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 — Recompute candidate upper slack with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute candidate lower slack with exact rational arithmetic, not rounded decimals.
- 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, -2], [-24, 25], -2, Yes, -14, evaluation_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: 8ed61f67a97fc7f5f41352a173a0976f6331aa08785935825a3068dfab332c4f
- candidate record SHA-256: 07de6fb234ae923494adee4bd214120deb9e0bc82326de9607478e226b11bed0
- semantic review receipt SHA-256: 94ec4350cf8b5e5c6c7701d536126eef20acb3a213e873ace4f5346d256b8fac
- question-first witness SHA-256: 0beba165ffd367a7c96de3aa58344d572d01434cd2ef772fbb00342c1f50b01b
- Lean: NOT_RUN · human review: NOT_PERFORMED

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