Question 22257: 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=2, B=-4, C=37, a=-8, b=9 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=-4, C=37, a=-8, b=9 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)=(2)t^2+(-4)t+(37);a=-8<b=9;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-16, 128, 32, 160, 197], [18, 162, -36, 126, 163]]

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(2, -8)

MVT endpointa ORIGINAL A times x = -16

MVT endpointa A*x times x := mul(-16, -8)

MVT endpointa A*x times x = 128

MVT endpointa ORIGINAL B times x := mul(-4, -8)

MVT endpointa ORIGINAL B times x = 32

MVT endpointa quadratic plus linear := sum(128, 32)

MVT endpointa quadratic plus linear = 160

MVT endpointa plus ORIGINAL constant := sum(160, 37)

MVT endpointa plus ORIGINAL constant = 197

MVT endpointb ORIGINAL A times x := mul(2, 9)

MVT endpointb ORIGINAL A times x = 18

MVT endpointb A*x times x := mul(18, 9)

MVT endpointb A*x times x = 162

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

MVT endpointb ORIGINAL B times x = -36

MVT endpointb quadratic plus linear := sum(162, -36)

MVT endpointb quadratic plus linear = 126

MVT endpointb plus ORIGINAL constant := sum(126, 37)

MVT endpointb plus ORIGINAL constant = 163

MVT ORIGINAL endpoint f(b) minus f(a) := sub(163, 197)

MVT ORIGINAL endpoint f(b) minus f(a) = -34

MVT ORIGINAL ordered interval length := sub(9, -8)

MVT ORIGINAL ordered interval length = 17

MVT exact secant slope := div(-34, 17)

MVT exact secant slope = -2

MVT ORIGINAL endpoint sum := sum(-8, 9)

MVT ORIGINAL endpoint sum = 1

MVT midpoint c := div(1, 2)

MVT midpoint c = 1/2

MVT c minus ORIGINAL a := sub(1/2, -8)

MVT c minus ORIGINAL a = 17/2

MVT ORIGINAL b minus c := sub(9, 1/2)

MVT ORIGINAL b minus c = 17/2

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

MVT derivative coefficient twice ORIGINAL A = 4

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

MVT derivative linear term at c = 2

MVT derivative plus ORIGINAL B := sum(2, -4)

MVT derivative plus ORIGINAL B = -2

ORIGINAL polynomial coefficients=[2, -4, 37];c=1/2,c-a=17/2>0,b-c=17/2>0

derivative2*A*c+B=-2;exact secant=-2;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, -2, -2, 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.