Suppose that the function  f: \mathbb {R} \rightarrow \mathbb {R} has two derivatives, with f(0)=f'(0)=0 and |f''(x)| \leq 1 if  |x| \leq 1 . Prove that f(x) \leq \frac {1}{2} if  |x| \leq 1

Proof so far.

Suppose that x \in \mathbb {R} with  |x| \leq 1 , pick x_0 \in \mathbb {R}, find z \in \mathbb {R} strictly between x and  x_0 such that f(x) = \frac {f''(z)}{2}(x-x_0)^2

Can I conclude that f''(z) \leq 1 ?