I am trying to learn measure theory by myself mostly following Folland. As I try to study I came up with the following question which I could not get an answer by myself.
Given a function $f\in L^2(\mathbb{R})$ with $f'\in L^2$, prove that
$$ \int \lvert f(x)\lvert^2 \leq 4 \int \lvert xf(x)\lvert^2 \int \lvert f'(x)\lvert ^2. $$
I guess the solution has something to do with Fourier transforms and it also states this inequality can be generalized to Heisenberg's uncertainty.
Hint: Observe \begin{align} \frac{d}{dx}|f(x)|^2 = 2f'(x)f(x) \end{align} then it follows \begin{align} \int |f(x)|^2\ dx = x|f(x)|^2\bigg|^\infty_{-\infty} - 2\int xf(x)f'(x)\ dx. \end{align} for any Schwartz function $f$.
Also, you probably forgot some square roots.