How to show that $\lim_{x\to \infty}f'(x)=0$

Question

Let $f$ be a real-valued, bounded, twice differentiable function defined on $(0,\infty)$ with $f'(x)\ge 0$ and $f''(x)\le 0$. Show that $$\lim_{x\to \infty}f'(x)=0$$

I understand $f: (0,\infty) \rightarrow \mathbb{R}, |f(x)|<M$ and $f''$ exists. But I coun't find the way to use the facts and show the limit is zero.

Answer 1

The limit of $f'$ exists because $f'$ is not increasing since $f''<0$. Suppose the limit is $m>0$. Then there exists $x_0$ such that for $x \ge x_0$ you would have $f'(x)\ge \frac m 2$ hence $f(x) \ge f(x_0) + \frac m 2 (x-x_0) \to +\infty$ as $x\to +\infty$.