Why is $f(x)g(x)$ a convex in $[a,b]$,when $f(x)$ and $g(x)$ are both convex and positive in $x \in [a,b ]$?

Question

Two non-increasing convex functions $f(x)$ and $g(x)$. If $f(x)$ and $g(x)$ are positive in $x \in [a,b ]$ , then f(x)g(x) is convex in $[a,b] $ Does anyone know how to prove this?

Answer 1

Hint:

$$(f(x)g(x))'' = f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x) \ge 0.$$