Consider an arbitrary polynomial function p(x) of degree n(n≥1). Answer the following:
a. (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. (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. (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.