Let's have two (real continuous differentiable) functions such that
- $f(x)$ is bounded (from below and from above), positive ($f(x)>0$), and is strictly increasing ($f'(x)>0$, $\forall x$).
- $g(x)$ is bounded (from below and from above) and has exactly one maximum ($g'(x_0) = 0$ ; $g(x)<g(x_0), \forall x \ne x_0$)
It then follows that function $h(x)=f(x)g(x)$ is also bounded. However, additional conditions on either $f(x)$ or $g(x)$ must apply in order for the function $h(x)$ to have again only one maximum.
The question is. What are the additional sufficient conditions on either $f(x)$ or $g(x)$ so that the product $h(x)=f(x)g(x)$ would have again only one maximum?
(Thanks to gammatester for pointing this out).