Question 1360: Coefficient Eligibility of a Candidate Antiderivative

Original question and givens

On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-4, 1], [-3, 1], [5, 1], [-1, 3], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?

• G001 — On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-4, 1], [-3, 1], [5, 1], [-1, 3], [3, 2]], 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=[-4, -3, 5, -1/3, 3/2]

target f coefficients=[-3, 10, -1, 2]

candidate derivative coefficient0 := mul(1, -3)

candidate derivative coefficient0 = -3

candidate derivative coefficient1 := mul(2, 5)

candidate derivative coefficient1 = 10

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

candidate derivative coefficient2 = -1

candidate derivative coefficient3 := mul(4, 3/2)

candidate derivative coefficient3 = 6

differentiate all positive-degree terms

F′ coefficients=[-3, 10, -1, 6]

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

F′ witness multiply at degree3 = 0

F′ witness add coefficient3 := sum(0, 6)

F′ witness add coefficient3 = 6

F′ witness multiply at degree2 := mul(6, 1)

F′ witness multiply at degree2 = 6

F′ witness add coefficient2 := sum(6, -1)

F′ witness add coefficient2 = 5

F′ witness multiply at degree1 := mul(5, 1)

F′ witness multiply at degree1 = 5

F′ witness add coefficient1 := sum(5, 10)

F′ witness add coefficient1 = 15

F′ witness multiply at degree0 := mul(15, 1)

F′ witness multiply at degree0 = 15

F′ witness add coefficient0 := sum(15, -3)

F′ witness add coefficient0 = 12

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

f witness multiply at degree3 = 0

f witness add coefficient3 := sum(0, 2)

f witness add coefficient3 = 2

f witness multiply at degree2 := mul(2, 1)

f witness multiply at degree2 = 2

f witness add coefficient2 := sum(2, -1)

f witness add coefficient2 = 1

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

f witness multiply at degree1 = 1

f witness add coefficient1 := sum(1, 10)

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, -3)

f witness add coefficient0 = 8

at x=1, F′(x)=12 and f(x)=8

one explicit mismatch refutes identity

F′ coefficients=[-3, 10, -1, 6]; f coefficients=[-3, 10, -1, 2]

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.