ISL 1970 Problem 25

A real function $f$ is defined for $0\le x\le 1$ , with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$ . Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$ , then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$ .