Here's a homework question I'm struggling with:
Prove/disprove the next statement:
Let $f,g$ two convex functions, then $h(x)=f(x) \cdot g(x)$ is also convex
So, we know that $h'(x)=f'(x) \cdot g(x) + f(x) \cdot g'(x)$. We also know that $f'(x),g'(x)$ are monotonically increasing because they are convex. If I can show that $h'(x)$ is also monotonically increasin I'm done, but I'm not sure how to do it. Any hints?