Question 22270: The Quadratic Midpoint Is a Strictly Interior MVT Witness

Original question and givens

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=9, C=362, a=-21, b=22 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=5, B=9, C=362, a=-21, b=22 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

Step-by-step derivation

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

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

ORIGINAL WHOLE f(t)=(5)t^2+(9)t+(362);a=-21<b=22;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-105, 2205, -189, 2016, 2378], [110, 2420, 198, 2618, 2980]]

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

MVT endpointa ORIGINAL A times x := mul(5, -21)

MVT endpointa ORIGINAL A times x = -105

MVT endpointa A*x times x := mul(-105, -21)

MVT endpointa A*x times x = 2205

MVT endpointa ORIGINAL B times x := mul(9, -21)

MVT endpointa ORIGINAL B times x = -189

MVT endpointa quadratic plus linear := sum(2205, -189)

MVT endpointa quadratic plus linear = 2016

MVT endpointa plus ORIGINAL constant := sum(2016, 362)

MVT endpointa plus ORIGINAL constant = 2378

MVT endpointb ORIGINAL A times x := mul(5, 22)

MVT endpointb ORIGINAL A times x = 110

MVT endpointb A*x times x := mul(110, 22)

MVT endpointb A*x times x = 2420

MVT endpointb ORIGINAL B times x := mul(9, 22)

MVT endpointb ORIGINAL B times x = 198

MVT endpointb quadratic plus linear := sum(2420, 198)

MVT endpointb quadratic plus linear = 2618

MVT endpointb plus ORIGINAL constant := sum(2618, 362)

MVT endpointb plus ORIGINAL constant = 2980

MVT ORIGINAL endpoint f(b) minus f(a) := sub(2980, 2378)

MVT ORIGINAL endpoint f(b) minus f(a) = 602

MVT ORIGINAL ordered interval length := sub(22, -21)

MVT ORIGINAL ordered interval length = 43

MVT exact secant slope := div(602, 43)

MVT exact secant slope = 14

MVT ORIGINAL endpoint sum := sum(-21, 22)

MVT ORIGINAL endpoint sum = 1

MVT midpoint c := div(1, 2)

MVT midpoint c = 1/2

MVT c minus ORIGINAL a := sub(1/2, -21)

MVT c minus ORIGINAL a = 43/2

MVT ORIGINAL b minus c := sub(22, 1/2)

MVT ORIGINAL b minus c = 43/2

MVT derivative coefficient twice ORIGINAL A := mul(2, 5)

MVT derivative coefficient twice ORIGINAL A = 10

MVT derivative linear term at c := mul(10, 1/2)

MVT derivative linear term at c = 5

MVT derivative plus ORIGINAL B := sum(5, 9)

MVT derivative plus ORIGINAL B = 14

ORIGINAL polynomial coefficients=[5, 9, 362];c=1/2,c-a=43/2>0,b-c=43/2>0

derivative2*A*c+B=14;exact secant=14;equal=Yes;A0 means ALL interior points work,not a unique midpoint

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, 14, 14, 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.