Prove $\int_{0}^{1} |\frac{f{''}(x)}{f(x)}|\,dx\ge 4$ when $f(0)=f(1)=0$.
If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$