Concavity and composite functions

Concavity and composite functions

Suppose U and g are two twice differentiable functions of x, both of them increasing and concave, with U’ ≥ 0, U” ≤ 0, g’ ≥ 0, and g” ≤ 0. Prove that the composite function f(x) = g(U(x)) is also increasing and concave
Concavity and composite functions