convex function with $f(0)=f'(0)=0$

Question

If $f\in C^2(\mathbb R)$ is strictly convex with $f(0)=0$ and $f'(0)=0$, is it true that $f(x)\geq 0$ for all $x\in\mathbb R$? I would say yes. $f$ is strictly convex so we have $f''(x)>0$, especially $f''(0)>0$ so $f$ has a minumum at $x=0$. My intuition says that $f$ cannot have a maximum and therefore not a second minimum. This implies that $f$ is $\geq 0$. Is this correct?