SCHWEITZER 1949 Problem 5

Let $f(x)$ be a polynomial of second degree the roots of which are contained in the interval $[-1,+1]$ and let there be a point $x_0\in [-1.+1]$ such that $|f(x_0)|=1$ . Prove that for every $\alpha \in [0,1]$ , there exists a $\zeta \in [-1,+1]$ such that $|f'(\zeta)|=\alpha$ and that this statement is not true if $\alpha>1$ .