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Let A, B ⊆R be intervals. Let f: A →R and g: B →R be differentiable...

Let A, B ⊆R be intervals. Let f: A →R and g: B →R be differentiable and such that f(A) ⊆ B. Recall that, by the Chain Rule, the composition g◦f: A →R is differentiable as well, and the formula
(g◦f)'(x) = g'(f(x))f'(x)
holds for all x ∈ A. Assume now that both f and g are twice differentiable.
(a) Prove that the composition g ◦ f is twice differentiable as well, and find a formula for the second derivative (g◦f)''.

(b) Prove that, if g is convex and f''(x) = 0 for all x ∈ A, then g◦f is convex.

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