Product of two functions with opposite sense of variation

Question

I have two functions, $f$ and $g$.

• $f<0$, strictly increasing, and bounded above (by $0$).
• $g<0$, strictly decreasing, and bounded above (by $0$).

Does $fg$ admit a maximum? Here's what I tried:

Let $h=fg$. Then $h'(x)= f'(x)g(x) + f(x)g'(x) = ``\leq 0" + ``\geq 0"$, which is inconclusive.