Question 22278: The Secant Uses Exact Endpoint Values and a Nonzero Denominator

Original question and givens

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=-8, C=5, a=-4, b=5 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 (value_at_a, value_at_b, interval_length, secant_slope, certificate).

• G001 — Use the whole real polynomial f(t)=A*t^2+B*t+C with A=3, B=-8, C=5, a=-4, b=5 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 (value_at_a, value_at_b, interval_length, secant_slope, 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-secant-slope;original snapshot and mathematics retained

explicit ordered source output fields=[value_at_a, value_at_b, interval_length, secant_slope, certificate]

ORIGINAL WHOLE f(t)=(3)t^2+(-8)t+(5);a=-4<b=5;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-12, 48, 32, 80, 85], [15, 75, -40, 35, 40]]

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(3, -4)

MVT endpointa ORIGINAL A times x = -12

MVT endpointa A*x times x := mul(-12, -4)

MVT endpointa A*x times x = 48

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

MVT endpointa ORIGINAL B times x = 32

MVT endpointa quadratic plus linear := sum(48, 32)

MVT endpointa quadratic plus linear = 80

MVT endpointa plus ORIGINAL constant := sum(80, 5)

MVT endpointa plus ORIGINAL constant = 85

MVT endpointb ORIGINAL A times x := mul(3, 5)

MVT endpointb ORIGINAL A times x = 15

MVT endpointb A*x times x := mul(15, 5)

MVT endpointb A*x times x = 75

MVT endpointb ORIGINAL B times x := mul(-8, 5)

MVT endpointb ORIGINAL B times x = -40

MVT endpointb quadratic plus linear := sum(75, -40)

MVT endpointb quadratic plus linear = 35

MVT endpointb plus ORIGINAL constant := sum(35, 5)

MVT endpointb plus ORIGINAL constant = 40

MVT ORIGINAL endpoint f(b) minus f(a) := sub(40, 85)

MVT ORIGINAL endpoint f(b) minus f(a) = -45

MVT ORIGINAL ordered interval length := sub(5, -4)

MVT ORIGINAL ordered interval length = 9

MVT exact secant slope := div(-45, 9)

MVT exact secant slope = -5

Reverse checks

• 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

(85, 40, 9, -5, exact_endpoint_secant_slope)

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.