OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 3651: Second Derivative of a Polynomial at a Point

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

Return to catalogue page 19

← Question derivation index · Original question batch

R0006-03651 · q-20261007-advance-20261009-r04-derivative-rules-0001

Original question and givens

On all R, f has ascending coefficients [0], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.

  • G001 — On all R, f has ascending coefficients [0], constant term first. Find f″(2). Use reduced positive-denominator fractions, retaining /1 for integers.

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.

Step-by-step derivation

S001 · implies

Before

f ascending coefficients=[0]

After

real polynomials are differentiable;use coefficient identities

Conditions: none

Coefficient k multiplies x^k;real polynomials differentiate termwise and constants have derivative0.Coefficient identities and a candidate comparison at one point are different claims.

Step identifiers and dependencies

Rule: polynomial_derivative_premise · Uses: G001

S002 · implies

Before

f″(x) ascending coefficients=[0]

After

evaluate at x=2

Conditions: none

Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.

Step identifiers and dependencies

Rule: polynomial_evaluation_order · Uses: G001, S001

S003 · implies

Before

first coefficients=[0];second=[0]

After

f″(2)=0

Conditions: none

Differentiate twice using each current degree;constants and linear polynomials have second derivative0.Do not square the first derivative.

Step identifiers and dependencies

Rule: successive_polynomial_derivatives · Uses: G001, S002

Reverse checks

  • 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 — 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

0/1

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 6669b46625ec87b01307a396ab8822ab1d31b78239e0e708cbc6fb1ac807a51b
  • candidate record SHA-256: 9412da7d4dcef674165b6327e0ccdd3c25b411610988d87706a9cf624c9038d2
  • semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
  • question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON