Prove the nth derivative test. Suppose n∈N, x0∈(a,b), and f:(a,b)→R is n times continuously differentiable, with f(k)(x0)=0 for k=1,2,…,n−1, and f(n)(x0)=0. Prove:
a) If n is odd, then f has neither a relative minimum, nor a maximum at x0.
b) If n is even, then f has a strict relative minimum at x0 if f(n)(x0)>0 and a strict relative maximum at x0 if f(n)(x0)<0.