Question 26380: Polynomial Continuity and Inclusive Endpoint Bracketing Are Separate Obligations

Original question and givens

Coefficients=[8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-9<b=-7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate).

• G001 — Coefficients=[8, 1] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Use a=-9<b=-7 and target=0 on the whole closed interval. Polynomial continuity and an inclusive endpoint-value bracket are separate hypotheses. Give (coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, 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-intermediate-value-theorem

Step-by-step derivation

source roles = [ivt-hypothesis]

derived = [[coefficients, a, b, value_at_a, value_at_b, target, polynomial_continuous, inclusive_bracket, certificate]]

(-9)^0

IVT ORIGINAL left endpoint ORIGINAL argument power0 = 1

IVT ORIGINAL left endpoint coefficient term0 := mul(8, 1)

IVT ORIGINAL left endpoint coefficient term0 = 8

IVT ORIGINAL left endpoint ascending partial sum0 := sum(0, 8)

IVT ORIGINAL left endpoint ascending partial sum0 = 8

(-9)^1

IVT ORIGINAL left endpoint ORIGINAL argument power1 = -9

IVT ORIGINAL left endpoint coefficient term1 := mul(1, -9)

IVT ORIGINAL left endpoint coefficient term1 = -9

IVT ORIGINAL left endpoint ascending partial sum1 := sum(8, -9)

IVT ORIGINAL left endpoint ascending partial sum1 = -1

(-7)^0

IVT ORIGINAL right endpoint ORIGINAL argument power0 = 1

IVT ORIGINAL right endpoint coefficient term0 := mul(8, 1)

IVT ORIGINAL right endpoint coefficient term0 = 8

IVT ORIGINAL right endpoint ascending partial sum0 := sum(0, 8)

IVT ORIGINAL right endpoint ascending partial sum0 = 8

(-7)^1

IVT ORIGINAL right endpoint ORIGINAL argument power1 = -7

IVT ORIGINAL right endpoint coefficient term1 := mul(1, -7)

IVT ORIGINAL right endpoint coefficient term1 = -7

IVT ORIGINAL right endpoint ascending partial sum1 := sum(8, -7)

IVT ORIGINAL right endpoint ascending partial sum1 = 1

source roles = [[8, 1], -9, -7, 0]

derived = [-1, 1, Yes, Yes]

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute IVT ORIGINAL right endpoint ascending partial sum1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute IVT ORIGINAL right endpoint coefficient term1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Recompute IVT ORIGINAL right endpoint ascending partial sum0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute IVT ORIGINAL right endpoint coefficient term0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute IVT ORIGINAL left endpoint ascending partial sum1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute IVT ORIGINAL left endpoint coefficient term1 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute IVT ORIGINAL left endpoint ascending partial sum0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute IVT ORIGINAL left endpoint coefficient term0 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

([8, 1], -9, -7, -1, 1, 0, Yes, Yes, polynomial_continuity_plus_inclusive_endpoint_bracket)

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.