References for this gradient inequality?

Question

I am looking for a proof or references to the following inequality. Let $f \in H^{1}(\mathbb{R}^{d})$ and $p = 2d/(d-2)$. Then: $$\|\nabla f\|_{L^{2}} \ge C\|f\|_{L^{p}}$$ for some constant $C > 0$ which is independent of $f$.

Any help is appreciated. Thanks!