¹⁶To see this, compute

d2 − dx2g(f(x))= − g′′f′− f-′′≥ − f′′ ddxg(f(x)) g′ f′ f′

where the inequality holds since $f′ ≥ 0$ , $g′ ≥ 0$ , and $g′′ ≤ 0$ .