BG-MO 1986 Problem 2

Let $f(x)$ be a quadratic polynomial with two real roots in the interval $[-1,1]$ . Prove that if the maximum value of $|f(x)|$ in the interval $[-1,1]$ is equal to $1$ , then the maximum value of $|f'(x)|$ in the interval $[-1,1]$ is not less than $1$ .