Let $f: [-1, 1] \longrightarrow [-1, 1]$ such that $f\in C^{1}$. Prove that there's exist $x_{0} \in [-1, 1]$ such that $|f'(x_{0})| \leq 1$ 
Let $f: [-1, 1] \longrightarrow [-1, 1]$ such that $f$ is a class $C^{1}$ function. Prove that there's exist $x_{0} \in [-1, 1]$ such that $|f'(x_{0})| \leq 1$.

I know that $f'([- 1,1])$ is compact, since $f'$ is continuous. Therefore, it is closed and limited. To prove the result, I tried to use the continuity of $f'$ in some sequence and tried to use the Weierstrass theorem, but I could not conclude anything. I would like some suggestion.

Note that
$$\frac{f(1)-f(-1)}{2} = f^\prime(\xi)$$
for some $\xi \in (-1,1)$
