If $f: (0, \infty)\to (0, \infty)$ is increasing, is it true that the function $x\longmapsto f'(x) \cdot x^2 $ is increasing? We can assume that $f$ is twice differentiable.
Can someone provide a counter-example, a function $f$ which is increasing and positive, but $f'(a)\cdot a^2 < f'(b)\cdot b^2$, for some $ a>b $ ?