1. Let I and J be intervals and suppose that the function f : I → J is twice differentiable on I. That is, the derivative f ′ exists and is itself differentiable on I. (We denote the derivative of f ′...

1. Let I and J be intervals and suppose that the function f : I → J is twice differentiable on I. That is, the derivative f ′ exists and is itself differentiable on I. (We denote the derivative of f ′ by f ″.) Suppose also that the function g : J →
 is twice differentiable on J. Prove that g ° f is twice differentiable on I and find (g ° f ) ″.

2. A function f :
 →
 is called an even function if f ( − x) = f (x) for all x ∈
. If f ( − x) = − f (x) for all x ∈ R, then f is called an odd function.

(a) Prove that if f is a differentiable even function, then f ′ is an odd function.

(b) Prove that if f is a differentiable odd function, then f ′ is an even function.