real analysis - When $ lim_{xrightarrow +infty}frac{f(x)}{x}=0$ implies $lim_{xrightarrow +infty}f'(x)=0$?

I have just solved a problem:

Let $f:[0,+\infty)\rightarrow \mathbb{R}$ be continuous on $[0,+\infty)$ and differentiable on $(0,+\infty)$. If $\displaystyle \lim_{x\rightarrow +\infty}f'(x)=0$, prove that $\displaystyle \lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0$

My questions is about the inverse: if $\displaystyle \lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0$, which hypothesis can be added (if needed) so that we can conclude $\displaystyle \lim_{x\rightarrow +\infty}f'(x)=0$ ?

Actually, I tried to solve the following one, but I still cannot solve it:

If $f:[0,+\infty)\rightarrow [0,+\infty)$ such that its second derivative is continuous, $f'\le 0$ and $|f''|\le M$ for some $M$ for all $x\ge 0$, then $\displaystyle \lim_{x\rightarrow +\infty}f'(x)=0$.

From the hypothesis, $f$ decreases and bounded below, so $f$ has a limit when $x$ tends to infinity. Thus $\displaystyle \lim_{x\rightarrow +\infty}\frac{f(x)}{x}=0$.

My questions seems to be in a wider range than the second problem above. Any help would be appreciated.