Product of linear and decreasing function has only one maxima

Question

I want to prove this statement: Suppose $g(x)$ is a decreasing function $\forall x>0$, hence $xg(x)$ has only one maxima $\forall x>0$.

Graphically it is clear that it has one maxima, but I need to prove it mathematically.

This question is similar to the one in Product of linear and convex function. I went through the answer, but I think it is not correct (when he gets the second derivative).

Can someone help?

Answer 1

Your statement is false as it is. Take $g(x) = \dfrac{1}{x^2}$. Then $g$ is decreasing for $x >0$, but $x g(x) = \dfrac{1}{x}$ is convex.

Answer 2

The argument is wrong, though it follows in many "non-artificial" scenarios. Here is a function which is a product of a linear and a strictly decreasing function. The product has two maxima. Use this google plot for visualising $$f(x)=x(e^{-0.5x}+e^{-50x^5})$$

PS: Late answer but might be useful for someone else.