Question 22235: Continuity and Differentiability Apply on Different Interval Parts

Original question and givens

Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=-1, C=82, a=-11, b=12 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 (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate).

• G001 — Use the whole real polynomial f(t)=A*t^2+B*t+C with A=5, B=-1, C=82, a=-11, b=12 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 (a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, 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-hypothesis-audit;original snapshot and mathematics retained

explicit ordered source output fields=[a, b, ordered_endpoints, closed_interval_continuity, open_interval_differentiability, certificate]

ORIGINAL WHOLE f(t)=(5)t^2+(-1)t+(82);a=-11<b=12;all endpoint[Ax,Axx,Bx,Axx+Bx,Axx+Bx+C] rows=[[-55, 605, 11, 616, 698], [60, 720, -12, 708, 790]]

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

Reverse checks

• 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

(-11, 12, Yes, Yes, Yes, polynomial_continuous_on_closed_interval_and_differentiable_on_open_interval)

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.