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?

score 3 · Accepted Answer · answered Nov 13 '18 at 15:53

3

Hint:

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

answered Nov 13 '18 at 15:53

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How do you show this when $f$ and $g$ aren't (twice) differentiable? – Theo Bendit Nov 13 '18 at 16:09
@TheoBendit https://math.stackexchange.com/questions/27571/convexity-of-the-product-of-two-functions-in-higher-dimensions – LinAlg Nov 13 '18 at 16:38

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

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