Let be defined for all real .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Let q(x)=x12+7x6q(x)=x^{12}+7x^{6} be defined for all real xx. (a) Determine the least positive integer nn for which q(n)(x)q^{(n)}(x) is identically zero for every real xx. Provide a clear argument supporting the minimality of your choice. (b) Prove that for any non-negative integer dd and any polynomial PP of degree dd, there exists an integer N≤d+1N\le d+1 such that P(N)(x)P^{(N)}(x) is identically zero for all real xx. Your proof should reference properties of repeated differentiation of power functions. (c) Explain how part (b) supports your conclusion in part (a), specifically for the coefficients and individual power terms present in q(x)q(x). Discuss why the term of highest degree controls the minimal nn you found.

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