Question 22284: 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=4, B=-2, C=65, a=-10, b=11 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=4, B=-2, C=65, a=-10, b=11 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)=(4)t^2+(-2)t+(65);a=-10<b=11;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-40, 400, 20, 420, 485], [44, 484, -22, 462, 527]]

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(4, -10)

MVT endpointa ORIGINAL A times x = -40

MVT endpointa A*x times x := mul(-40, -10)

MVT endpointa A*x times x = 400

MVT endpointa ORIGINAL B times x := mul(-2, -10)

MVT endpointa ORIGINAL B times x = 20

MVT endpointa quadratic plus linear := sum(400, 20)

MVT endpointa quadratic plus linear = 420

MVT endpointa plus ORIGINAL constant := sum(420, 65)

MVT endpointa plus ORIGINAL constant = 485

MVT endpointb ORIGINAL A times x := mul(4, 11)

MVT endpointb ORIGINAL A times x = 44

MVT endpointb A*x times x := mul(44, 11)

MVT endpointb A*x times x = 484

MVT endpointb ORIGINAL B times x := mul(-2, 11)

MVT endpointb ORIGINAL B times x = -22

MVT endpointb quadratic plus linear := sum(484, -22)

MVT endpointb quadratic plus linear = 462

MVT endpointb plus ORIGINAL constant := sum(462, 65)

MVT endpointb plus ORIGINAL constant = 527

MVT ORIGINAL endpoint f(b) minus f(a) := sub(527, 485)

MVT ORIGINAL endpoint f(b) minus f(a) = 42

MVT ORIGINAL ordered interval length := sub(11, -10)

MVT ORIGINAL ordered interval length = 21

MVT exact secant slope := div(42, 21)

MVT exact secant slope = 2

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

(485, 527, 21, 2, 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.