# Question 22252: The Quadratic Midpoint Is a Strictly Interior MVT Witness

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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.

R0043-22252 · q-20261014-advance-20261110-r02-mean-value-theorem-0052



## Original question and givens

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=-9, C=2, a=-3, b=4 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).


- G001 — Use the whole real polynomial f(t)=A*t^2+B*t+C with A=2, B=-9, C=2, a=-3, b=4 and a<b. Coefficients are quadratic, linear and constant in that order. This declared polynomial is continuous on [a,b] and differentiable on (a,b); finite samples do not certify those hypotheses for an arbitrary function. Give (c, strictly_interior, derivative_at_c, secant_slope, equal, certificate).





## 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-mean-value-theorem](https://ocme.evemisslab.com/en/mko/mko-mean-value-theorem/)





## Step-by-step derivation

S001 · impliesBefore

source operation=mvt-polynomial-witness;original snapshot and mathematics retained

After

explicit ordered source output fields=[c, strictly_interior, derivative_at_c, secant_slope, equal, certificate]

Conditions: none

Retain the historical interface marker and list ordered source fields without JavaScript.MVT fields come from the original mode,without padding or answer-shape inference.Tags and fixed No markers retain their claim scope.

Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001





S002 · impliesBefore

ORIGINAL WHOLE f(t)=(2)t^2+(-9)t+(2);a=-3<b=4;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-6, 18, 27, 45, 47], [8, 32, -36, -4, -2]]

After

English source ALL endpoint intermediate SAFE guards passed;polynomial continuous on CLOSED[a,b] and differentiable on OPEN(a,b),not inferred from finite samples

Conditions: none

Finite sums/products preserve continuity and differentiability,with derivative2At+B.These are properties of the whole formula,not an inference from endpoint samples.SAFE guards qualify source numeric projection,not the mathematical domain of polynomials.

Step identifiers and dependenciesRule: mvt_whole_polynomial_hypotheses_and_SAFE_source_guard · Uses: G001, S001





S003 · calculationBefore

MVT endpointa ORIGINAL A times x := mul(2, -3)

After

MVT endpointa ORIGINAL A times x = -6

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

MVT endpointa A*x times x := mul(-6, -3)

After

MVT endpointa A*x times x = 18

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

MVT endpointa ORIGINAL B times x := mul(-9, -3)

After

MVT endpointa ORIGINAL B times x = 27

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

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





S006 · calculationBefore

MVT endpointa quadratic plus linear := sum(18, 27)

After

MVT endpointa quadratic plus linear = 45

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

MVT endpointa plus ORIGINAL constant := sum(45, 2)

After

MVT endpointa plus ORIGINAL constant = 47

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

MVT endpointb ORIGINAL A times x := mul(2, 4)

After

MVT endpointb ORIGINAL A times x = 8

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · calculationBefore

MVT endpointb A*x times x := mul(8, 4)

After

MVT endpointb A*x times x = 32

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008





S010 · calculationBefore

MVT endpointb ORIGINAL B times x := mul(-9, 4)

After

MVT endpointb ORIGINAL B times x = -36

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

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





S011 · calculationBefore

MVT endpointb quadratic plus linear := sum(32, -36)

After

MVT endpointb quadratic plus linear = -4

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

MVT endpointb plus ORIGINAL constant := sum(-4, 2)

After

MVT endpointb plus ORIGINAL constant = -2

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

MVT ORIGINAL endpoint f(b) minus f(a) := sub(-2, 47)

After

MVT ORIGINAL endpoint f(b) minus f(a) = -49

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

MVT ORIGINAL ordered interval length := sub(4, -3)

After

MVT ORIGINAL ordered interval length = 7

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · calculationBefore

MVT exact secant slope := div(-49, 7)

After

MVT exact secant slope = -7

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014





S016 · calculationBefore

MVT ORIGINAL endpoint sum := sum(-3, 4)

After

MVT ORIGINAL endpoint sum = 1

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

MVT midpoint c := div(1, 2)

After

MVT midpoint c = 1/2

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

MVT c minus ORIGINAL a := sub(1/2, -3)

After

MVT c minus ORIGINAL a = 7/2

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · calculationBefore

MVT ORIGINAL b minus c := sub(4, 1/2)

After

MVT ORIGINAL b minus c = 7/2

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

MVT derivative coefficient twice ORIGINAL A := mul(2, 2)

After

MVT derivative coefficient twice ORIGINAL A = 4

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · calculationBefore

MVT derivative linear term at c := mul(4, 1/2)

After

MVT derivative linear term at c = 2

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020





S022 · calculationBefore

MVT derivative plus ORIGINAL B := sum(2, -9)

After

MVT derivative plus ORIGINAL B = -7

Conditions: none

Substitute original roles,order and established intermediates exactly,without changing source bindings or using floating-point approximations.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · impliesBefore

ORIGINAL polynomial coefficients=[2, -9, 2];c=1/2,c-a=7/2>0,b-c=7/2>0

After

derivative2*A*c+B=-7;exact secant=-7;equal=Yes;A0 means ALL interior points work,not a unique midpoint

Conditions: none

Algebraically f(b)−f(a)=(b−a)(A(a+b)+B),so the midpoint derivative equals the secant.For general existence,subtract the secant affine function to obtain equal endpoints;Rolle follows from a closed-interval extremum and zero derivative at an interior extremum.This is not a sampling proof,and A=0 does not yield uniqueness.

Step identifiers and dependenciesRule: mvt_strict_interior_derivative_equals_secant_not_unique · Uses: G001, S022









## Reverse checks


- S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S022 · premise_and_result_audit — Recompute MVT derivative plus ORIGINAL B with exact rational arithmetic, not rounded decimals.
- S021 · premise_and_result_audit — Recompute MVT derivative linear term at c with exact rational arithmetic, not rounded decimals.
- S020 · premise_and_result_audit — Recompute MVT derivative coefficient twice ORIGINAL A with exact rational arithmetic, not rounded decimals.
- S019 · premise_and_result_audit — Recompute MVT ORIGINAL b minus c with exact rational arithmetic, not rounded decimals.
- S018 · premise_and_result_audit — Recompute MVT c minus ORIGINAL a with exact rational arithmetic, not rounded decimals.
- S017 · premise_and_result_audit — Recompute MVT midpoint c with exact rational arithmetic, not rounded decimals.
- S016 · premise_and_result_audit — Recompute MVT ORIGINAL endpoint sum with exact rational arithmetic, not rounded decimals.
- S015 · premise_and_result_audit — Recompute MVT exact secant slope with exact rational arithmetic, not rounded decimals.
- S014 · premise_and_result_audit — Recompute MVT ORIGINAL ordered interval length with exact rational arithmetic, not rounded decimals.
- S013 · premise_and_result_audit — Recompute MVT ORIGINAL endpoint f(b) minus f(a) with exact rational arithmetic, not rounded decimals.
- S012 · premise_and_result_audit — Recompute MVT endpointb plus ORIGINAL constant with exact rational arithmetic, not rounded decimals.
- S011 · premise_and_result_audit — Recompute MVT endpointb quadratic plus linear with exact rational arithmetic, not rounded decimals.
- S010 · premise_and_result_audit — Recompute MVT endpointb ORIGINAL B times x with exact rational arithmetic, not rounded decimals.
- S009 · premise_and_result_audit — Recompute MVT endpointb A*x times x with exact rational arithmetic, not rounded decimals.
- S008 · premise_and_result_audit — Recompute MVT endpointb ORIGINAL A times x with exact rational arithmetic, not rounded decimals.
- S007 · premise_and_result_audit — Recompute MVT endpointa plus ORIGINAL constant with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute MVT endpointa quadratic plus linear with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute MVT endpointa ORIGINAL B times x with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute MVT endpointa A*x times x with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute MVT endpointa ORIGINAL A times x 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

(1/2, Yes, -7, -7, Yes, quadratic_midpoint_witness)

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: e980393e1a9bc336496e008ae4581e18ee78d896f7ff5423849314c3e2809b43
- candidate record SHA-256: bf325576adea88fd48421184f012cdf377cbb7f60f8114fbdedb8c9a3cefd81c
- semantic review receipt SHA-256: 49cf6ec60211c92d1c0c302663af4d44479635dc8181fb61a20c3374288e21c5
- question-first witness SHA-256: 10fd25b4643e5ff12753daceaa61f9f2499bf381f65ef96e9d21185c7c0d0c5e
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r02-mean-value-theorem-0052.json)