Question 22294: 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=8, C=325, a=-20, b=21 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=8, C=325, a=-20, b=21 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+(8)t+(325);a=-20<b=21;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-80, 1600, -160, 1440, 1765], [84, 1764, 168, 1932, 2257]]

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, -20)

MVT endpointa ORIGINAL A times x = -80

MVT endpointa A*x times x := mul(-80, -20)

MVT endpointa A*x times x = 1600

MVT endpointa ORIGINAL B times x := mul(8, -20)

MVT endpointa ORIGINAL B times x = -160

MVT endpointa quadratic plus linear := sum(1600, -160)

MVT endpointa quadratic plus linear = 1440

MVT endpointa plus ORIGINAL constant := sum(1440, 325)

MVT endpointa plus ORIGINAL constant = 1765

MVT endpointb ORIGINAL A times x := mul(4, 21)

MVT endpointb ORIGINAL A times x = 84

MVT endpointb A*x times x := mul(84, 21)

MVT endpointb A*x times x = 1764

MVT endpointb ORIGINAL B times x := mul(8, 21)

MVT endpointb ORIGINAL B times x = 168

MVT endpointb quadratic plus linear := sum(1764, 168)

MVT endpointb quadratic plus linear = 1932

MVT endpointb plus ORIGINAL constant := sum(1932, 325)

MVT endpointb plus ORIGINAL constant = 2257

MVT ORIGINAL endpoint f(b) minus f(a) := sub(2257, 1765)

MVT ORIGINAL endpoint f(b) minus f(a) = 492

MVT ORIGINAL ordered interval length := sub(21, -20)

MVT ORIGINAL ordered interval length = 41

MVT exact secant slope := div(492, 41)

MVT exact secant slope = 12

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

(1765, 2257, 41, 12, 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.