$f$ real-valued, continuously differentiable, then $(\int |f|^2\,dx)^2 \le 4(\int |xf(x)|^2\,dx)(\int |f'|^2\,dx)$?
Prove an integral inequality: $ \left(\int|f|^2dx\right)^2\le 4\left(\int|xf(x)|^2dx\right)\left(\int|f'|^2dx\right) $