¹⁴To see this, compute

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

(20)

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