Question 1354: Coefficient Eligibility of a Candidate Antiderivative

Original question and givens

On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-10, 1], [-9, 1], [2, 1], [-1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?

• G001 — On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-10, 1], [-9, 1], [2, 1], [-1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?

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-antiderivative

Step-by-step derivation

domain=R; coefficient lists are in ascending powers

constant term first; rational pairs=[numerator,denominator]

candidate F coefficients=[-10, -9, 2, -1/3, 2]

target f coefficients=[-9, 4, -1, 4]

candidate derivative coefficient0 := mul(1, -9)

candidate derivative coefficient0 = -9

candidate derivative coefficient1 := mul(2, 2)

candidate derivative coefficient1 = 4

candidate derivative coefficient2 := mul(3, -1/3)

candidate derivative coefficient2 = -1

candidate derivative coefficient3 := mul(4, 2)

candidate derivative coefficient3 = 8

differentiate all positive-degree terms

F′ coefficients=[-9, 4, -1, 8]

F′ witness multiply at degree3 := mul(0, 1)

F′ witness multiply at degree3 = 0

F′ witness add coefficient3 := sum(0, 8)

F′ witness add coefficient3 = 8

F′ witness multiply at degree2 := mul(8, 1)

F′ witness multiply at degree2 = 8

F′ witness add coefficient2 := sum(8, -1)

F′ witness add coefficient2 = 7

F′ witness multiply at degree1 := mul(7, 1)

F′ witness multiply at degree1 = 7

F′ witness add coefficient1 := sum(7, 4)

F′ witness add coefficient1 = 11

F′ witness multiply at degree0 := mul(11, 1)

F′ witness multiply at degree0 = 11

F′ witness add coefficient0 := sum(11, -9)

F′ witness add coefficient0 = 2

f witness multiply at degree3 := mul(0, 1)

f witness multiply at degree3 = 0

f witness add coefficient3 := sum(0, 4)

f witness add coefficient3 = 4

f witness multiply at degree2 := mul(4, 1)

f witness multiply at degree2 = 4

f witness add coefficient2 := sum(4, -1)

f witness add coefficient2 = 3

f witness multiply at degree1 := mul(3, 1)

f witness multiply at degree1 = 3

f witness add coefficient1 := sum(3, 4)

f witness add coefficient1 = 7

f witness multiply at degree0 := mul(7, 1)

f witness multiply at degree0 = 7

f witness add coefficient0 := sum(7, -9)

f witness add coefficient0 = -2

at x=1, F′(x)=2 and f(x)=-2

one explicit mismatch refutes identity

F′ coefficients=[-9, 4, -1, 8]; f coefficients=[-9, 4, -1, 4]

F′=f identically: No

Reverse checks

• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Recompute f witness add coefficient0 with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute f witness multiply at degree0 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute f witness add coefficient1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute f witness multiply at degree1 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute f witness add coefficient2 with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute f witness multiply at degree2 with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute f witness add coefficient3 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute f witness multiply at degree3 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute F′ witness add coefficient0 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute F′ witness multiply at degree0 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute F′ witness add coefficient1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute F′ witness multiply at degree1 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute F′ witness add coefficient2 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute F′ witness multiply at degree2 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute F′ witness add coefficient3 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute F′ witness multiply at degree3 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute candidate derivative coefficient3 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute candidate derivative coefficient2 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute candidate derivative coefficient1 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute candidate derivative coefficient0 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

No

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.