Find $M_{min}$ if there exist constant $M$ such $f(x)If   $\dfrac{f(x)}{x^2}$monotone increasing function on $x\in (0,+\infty)$,and there exist constant $M$,such $f(x)<M,\forall x\in (0,+\infty)$,then Find the $M_{min}$
If we let $g(x)=\dfrac{f(x)}{x^2}$,then for any $x,y>0(x<y)$,we have $g(x)<g(y)$ or
$$\dfrac{f(x)}{x^2}<\dfrac{f(y)}{y^2}$$
but I don't have any idea how to start proving it,
Thanks

The function $f$ must not have an infimum. Take $f(x) := -\frac{1}{x}$, then $f(x)$ is bounded from above by $0$ and $\frac{f(x)}{x^2} = -\frac{1}{x^3}$ is monotonically increasing, but $f(x) \to - \infty$ as $x \to 0$.
