Question 26431: A Constant Next Derivative Gives a Strictly Intermediate Lagrange Witness

Original question and givens

Coefficients=[7, 13, 7] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate).

• G001 — Coefficients=[7, 13, 7] are in ascending power order, with exact rationals and trailing zero coefficients removed for degree. Center=0, point=1/2, order=1; keep terms through that order, and define remainder=f(x)−Taylor_value. Give (degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, 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-taylor-remainder-theorem

Step-by-step derivation

source roles = [taylor-lagrange-witness]

derived = [[degree, order, constant_next_derivative, intermediate_point, strictly_between, Lagrange_remainder, actual_remainder, remainders_equal, certificate]]

Taylor ORIGINAL x minus declared center := sub(1/2, 0)

Taylor ORIGINAL x minus declared center = 1/2

(0)^0

Taylor ORIGINAL derivative0 center power0 term0 = 1

Taylor ORIGINAL derivative0 falling factor times power term0 := mul(1, 1)

Taylor ORIGINAL derivative0 falling factor times power term0 = 1

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 := mul(7, 1)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 = 7

Taylor ORIGINAL derivative0 ascending derivative sum term0 := sum(0, 7)

Taylor ORIGINAL derivative0 ascending derivative sum term0 = 7

(0)^1

Taylor ORIGINAL derivative0 center power1 term1 = 0

Taylor ORIGINAL derivative0 falling factor times power term1 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term1 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 := mul(13, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term1 := sum(7, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term1 = 7

(0)^2

Taylor ORIGINAL derivative0 center power2 term2 = 0

Taylor ORIGINAL derivative0 falling factor times power term2 := mul(1, 0)

Taylor ORIGINAL derivative0 falling factor times power term2 = 0

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 := mul(7, 0)

Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative0 ascending derivative sum term2 := sum(7, 0)

Taylor ORIGINAL derivative0 ascending derivative sum term2 = 7

source roles = [0, 0, [7, 13, 7]]

derived = [[[0, 7, 1, 1, 7], [1, 13, 1, 0, 0], [2, 7, 1, 0, 0]], 7]

(1/2)^0

Taylor ORIGINAL displacement power0 = 1

Taylor ORIGINAL derivative times displacement0 := mul(7, 1)

Taylor ORIGINAL derivative times displacement0 = 7

Taylor ORIGINAL term0 divide factorial := div(7, 1)

Taylor ORIGINAL term0 divide factorial = 7

(0)^0

Taylor ORIGINAL derivative1 center power0 term1 = 1

Taylor ORIGINAL derivative1 falling factor times power term1 := mul(1, 1)

Taylor ORIGINAL derivative1 falling factor times power term1 = 1

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 := mul(13, 1)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 = 13

Taylor ORIGINAL derivative1 ascending derivative sum term1 := sum(0, 13)

Taylor ORIGINAL derivative1 ascending derivative sum term1 = 13

(0)^1

Taylor ORIGINAL derivative1 center power1 term2 = 0

Taylor ORIGINAL derivative1 falling factor times power term2 := mul(2, 0)

Taylor ORIGINAL derivative1 falling factor times power term2 = 0

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 := mul(7, 0)

Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 = 0

Taylor ORIGINAL derivative1 ascending derivative sum term2 := sum(13, 0)

Taylor ORIGINAL derivative1 ascending derivative sum term2 = 13

source roles = [1, 0, [7, 13, 7]]

derived = [[[1, 13, 1, 1, 13], [2, 7, 2, 0, 0]], 13]

(1/2)^1

Taylor ORIGINAL displacement power1 = 1/2

Taylor ORIGINAL derivative times displacement1 := mul(13, 1/2)

Taylor ORIGINAL derivative times displacement1 = 13/2

Taylor ORIGINAL term1 divide factorial := div(13/2, 1)

Taylor ORIGINAL term1 divide factorial = 13/2

Taylor ORIGINAL partial sum0 := sum(0, 7)

Taylor ORIGINAL partial sum0 = 7

Taylor ORIGINAL partial sum1 := sum(7, 13/2)

Taylor ORIGINAL partial sum1 = 27/2

(1/2)^0

Taylor ORIGINAL full function at x ORIGINAL argument power0 = 1

Taylor ORIGINAL full function at x coefficient term0 := mul(7, 1)

Taylor ORIGINAL full function at x coefficient term0 = 7

Taylor ORIGINAL full function at x ascending partial sum0 := sum(0, 7)

Taylor ORIGINAL full function at x ascending partial sum0 = 7

(1/2)^1

Taylor ORIGINAL full function at x ORIGINAL argument power1 = 1/2

Taylor ORIGINAL full function at x coefficient term1 := mul(13, 1/2)

Taylor ORIGINAL full function at x coefficient term1 = 13/2

Taylor ORIGINAL full function at x ascending partial sum1 := sum(7, 13/2)

Taylor ORIGINAL full function at x ascending partial sum1 = 27/2

(1/2)^2

Taylor ORIGINAL full function at x ORIGINAL argument power2 = 1/4

Taylor ORIGINAL full function at x coefficient term2 := mul(7, 1/4)

Taylor ORIGINAL full function at x coefficient term2 = 7/4

Taylor ORIGINAL full function at x ascending partial sum2 := sum(27/2, 7/4)

Taylor ORIGINAL full function at x ascending partial sum2 = 61/4

Taylor ORIGINAL f(x) minus Taylor value := sub(61/4, 27/2)

Taylor ORIGINAL f(x) minus Taylor value = 7/4

source roles = [[7, 13, 7], 0, 1/2, 1, [7, 13], [7, 13/2]]

derived = [27/2, 61/4, 7/4, 2]

(0)^0

Taylor ORIGINAL next derivative center power0 term2 = 1

Taylor ORIGINAL next derivative falling factor times power term2 := mul(2, 1)

Taylor ORIGINAL next derivative falling factor times power term2 = 2

Taylor ORIGINAL next derivative ORIGINAL coefficient term2 := mul(7, 2)

Taylor ORIGINAL next derivative ORIGINAL coefficient term2 = 14

Taylor ORIGINAL next derivative ascending derivative sum term2 := sum(0, 14)

Taylor ORIGINAL next derivative ascending derivative sum term2 = 14

source roles = [2, 0, [7, 13, 7]]

derived = [[[2, 7, 2, 1, 14]], 14]

Taylor witness ORIGINAL center plus x := sum(0, 1/2)

Taylor witness ORIGINAL center plus x = 1/2

Taylor witness ORIGINAL midpoint := div(1/2, 2)

Taylor witness ORIGINAL midpoint = 1/4

(1/2)^2

Taylor witness ORIGINAL displacement power = 1/4

Taylor witness constant derivative times power := mul(14, 1/4)

Taylor witness constant derivative times power = 7/2

Taylor witness divide next factorial := div(7/2, 2)

Taylor witness divide next factorial = 7/4

source roles = [2, 1, 0, 1/2, 14, 1/4]

derived = [Yes, 7/4, 7/4, Yes]

Reverse checks

• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Recompute Taylor witness divide next factorial with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Taylor witness constant derivative times power with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Recompute Taylor witness ORIGINAL midpoint with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Taylor witness ORIGINAL center plus x with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Recompute Taylor ORIGINAL next derivative ascending derivative sum term2 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Taylor ORIGINAL next derivative ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Recompute Taylor ORIGINAL next derivative falling factor times power term2 with exact rational arithmetic, not rounded decimals.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S042 · premise_and_result_audit — Recompute Taylor ORIGINAL f(x) minus Taylor value with exact rational arithmetic, not rounded decimals.
• S041 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum2 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term2 with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S038 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum1 with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term1 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x ascending partial sum0 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Taylor ORIGINAL full function at x coefficient term0 with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S032 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum1 with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Taylor ORIGINAL partial sum0 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Taylor ORIGINAL term1 divide factorial with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement1 with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S026 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term2 with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ascending derivative sum term1 with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 ORIGINAL coefficient term1 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative1 falling factor times power term1 with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute Taylor ORIGINAL term0 divide factorial with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative times displacement0 with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ascending derivative sum term2 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term2 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term2 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 Taylor ORIGINAL derivative0 ascending derivative sum term1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term1 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term1 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 Taylor ORIGINAL derivative0 ascending derivative sum term0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 ORIGINAL coefficient term0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Taylor ORIGINAL derivative0 falling factor times power term0 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute Taylor ORIGINAL x minus declared center with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(2, 1, 14, 1/4, Yes, 7/4, 7/4, Yes, constant_n_plus_one_derivative_gives_exact_strictly_intermediate_Lagrange_witness)

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.