Twice differentiable continuous function estimate

Suppose that a function $\displaystyle f: \mathbb {R} \rightarrow \mathbb {R} $ is twice differentiable and both $\displaystyle f', f'' $ are continuous, also suppose that $\displaystyle f(0)=f'(0)=0$, show that $\displaystyle \exists c>0$ such that $\displaystyle |f(x)| \leq cx^2 $.

Proof so far.

By the Taylor's Theorem, we know that $\displaystyle f(x) = f(0)+f'(0)x+ \frac {1}{2} f''( \epsilon ) x^2 $ where $\displaystyle \epsilon $ is a number between 0 and x.

So we have $\displaystyle f(x) = \frac {1}{2} f''( \epsilon ) x^2 $

Let $\displaystyle c= \frac {1}{2}f''( \epsilon ) $, then $\displaystyle |f(x)| \leq cx^2 $.

Is this right? Thanks.