Prove that $\int_0^1 \left| \frac{f''(x)}{f(x)} \right| dx \geq 4$, under the given conditions on $f(x)$

Suppose $f(x) \in C^2[0,1]$ such that $f(0)=f(1)=0$, and $f(x)\neq 0$ on $(0,1)$. Prove that
$$\int_0^1 \left| \frac{f''(x)}{f(x)} \right| dx \geq 4.$$


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