Monotony of product of functions

Question

Is there any theorem that states something about the monotony of a product of functions? Let's say $f$ and $g$ are stictly increasing on $\mathbb R$. Does this mean that $f\cdot g$ is strictly increasing on $\mathbb R$ ? If case it's not, what if $f$ and $g$ are positive (or negative) on $\mathbb R$ ?