Consider an arbitrary polynomial function of degree
MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions
Problem
Consider an arbitrary polynomial function p(x) of degree n(n≥1). Answer the following:
(a) Define the "k-th derivative" of a function for a positive integer k. State under what condition two successive derivatives of a polynomial can be identical.
(b) Prove that the (n+1)-th derivative of p(x) is identically zero for every x∈R. (You may assume basic results on limits and differentiation.)
(c) Hence or otherwise, explain why the highest non-zero derivative of a polynomial of degree n is its n-th derivative. Illustrate your explanation with the example p(x)=5x4−7x2+3. List each non-zero derivative and indicate at which order the derivatives become zero.