Applying a Wirtinger-like inequality to get a bound for a differential equation.

Here I am considering the following problem.

I have some $f:[0,\infty) \to \mathbb R$ smooth with the property that $f''≥-Cf$ for some $C > 0$.

We have that $f(0)=0$ but $f'(0)>0$.

Can I find an $R>0$ just determined by $C$ such that $f>0$ on $(0,R)$?

I have the following hint:

'The following variant of Wirtinger's inequality may be useful and can be assumed without proof: if $g$ is a $C^1$ function on $[0, L]$ vanishing at $0$ , then $\int_0^L|g(x)|^2 d x \leqslant \frac{L}{2 \pi} \int_0^L\left|g^{\prime}(x)\right|^2 d x$.'

It is not clear to me how to apply it: all of the bounds I get end up being self-referential and not that helpful here.