How to prove: $\int\limits_0^1 \frac{|f''(x)|}{f(x)}dx>4,$ using elementary math tools.

Question

Let $f\in C^2 [0,1]$ and $f(0)=0=f(1)$ and for all $x\in (0,1)$ we have $f(x)>0$. Prove that

$$ \int_0^1 \dfrac{|f''(x)|}{f(x)}dx>4.$$

I want to prove this problem with elementary math tools. It is given to the first year university students as a calculus exercise. It is noticed that:

Hint: Use average value theorem in suitable intervals.