Question 22290: 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=5, B=4, C=197, a=-16, b=17 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=5, B=4, C=197, a=-16, b=17 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)=(5)t^2+(4)t+(197);a=-16<b=17;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-80, 1280, -64, 1216, 1413], [85, 1445, 68, 1513, 1710]]

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, -16)

MVT endpointa ORIGINAL A times x = -80

MVT endpointa A*x times x := mul(-80, -16)

MVT endpointa A*x times x = 1280

MVT endpointa ORIGINAL B times x := mul(4, -16)

MVT endpointa ORIGINAL B times x = -64

MVT endpointa quadratic plus linear := sum(1280, -64)

MVT endpointa quadratic plus linear = 1216

MVT endpointa plus ORIGINAL constant := sum(1216, 197)

MVT endpointa plus ORIGINAL constant = 1413

MVT endpointb ORIGINAL A times x := mul(5, 17)

MVT endpointb ORIGINAL A times x = 85

MVT endpointb A*x times x := mul(85, 17)

MVT endpointb A*x times x = 1445

MVT endpointb ORIGINAL B times x := mul(4, 17)

MVT endpointb ORIGINAL B times x = 68

MVT endpointb quadratic plus linear := sum(1445, 68)

MVT endpointb quadratic plus linear = 1513

MVT endpointb plus ORIGINAL constant := sum(1513, 197)

MVT endpointb plus ORIGINAL constant = 1710

MVT ORIGINAL endpoint f(b) minus f(a) := sub(1710, 1413)

MVT ORIGINAL endpoint f(b) minus f(a) = 297

MVT ORIGINAL ordered interval length := sub(17, -16)

MVT ORIGINAL ordered interval length = 33

MVT exact secant slope := div(297, 33)

MVT exact secant slope = 9

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

(1413, 1710, 33, 9, 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.