I am doing a question below
Let $f\in C^{1},~2\pi~periodic~and~f(x_{0})=0~for~some~x_{0}\in~[-\pi,\pi]~and$ $$\frac{1}{2\pi}\int_{\pi}^{\pi}|f(x)|^{2}dx=1$$ then show $$(\sum n^{2}|\hat{f}(n)|^{2})\int_{-\pi}^{\pi}(x-x_{0})^2f^{2}(x)dx > \frac{\pi}{2}$$ where $\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$
My idea is to first prove the case that $x_{0}=0$ by using a method similar to proving Heisenberg's uncertainty principle (refer to Stein's Fourier Analysis chapter 5), but I am stuck in this part because I don't know how to use the condition $f(x_{0})=0$. Any help is appreciated!