Let $f \in L^{2}(\mathbb{R})$ be continuously differentiable on $\mathbb{R}$. I am trying to show the following: $( \int |f|^{2} dx)^{2} \leq 4 ( \int |xf(x)|^{2} dx) ( \int |f'|^{2} dx))$.
My first thought is to think about this inequality as $ ||f||^{4}_{2} \leq 4 ||xf(x)||^{2}_{2} ||f'||^{2}_{2} $ and apply Holder's inequality to get $ 4 (\int |xf(x)f'(x)| dx)^{2} \leq 4 ||xf(x)||^{2}_{2} ||f'||^{2}_{2}$ but beyond this I have no intuition, especially what to do with the continuously differentiable assumption. Could someone lend me a hint?