If $f(x)>0$ is increasing and $g(x)>0$ is decreasing, then, is it clear if $f(x)g(x)$ is increasing or decreasing?

Question

If $f(x)>0$ and increasing for $x>0$, and $g(x)>0$ and decreasing for $x>0$, then, is it clear if $f(x)g(x)$ is increasing or decreasing?

Answer 1

No, consider

$$f(x)=x^2$$

$$g(x)=1/x$$

Then $fg=x$ is increasing. However

$$f(x)=x$$

$$g(x)=1/x^2$$

then $fg=1/x$ is decreasing.

Answer 2

No it is not clear.

There can be some parts in which $f$ is almost constant while $g$ decreases fast.

There can be some parts in which $f$ increases fast while $g$ is almost constant.

Answer 3

Take $f(x) = x$ and $g(x) = 1/x$.