I'm trying to prove the following inequality $$\left\||x|f\right\|_{L^\infty(\mathbb{R}^3)}\lesssim \left\|f\right\|_{H^1(\mathbb{R}^3)},$$ for every radial function $f\in H^1(\mathbb{R}^3)$, where $\left\|f\right\|_{H^1(\mathbb{R}^3)}:=\left\|f\right\|_{L^2(\mathbb{R}^3)}+\left\|\nabla f\right\|_{L^2(\mathbb{R}^3)}$. There's a hint that tells one should use the fundamental theorem of calculus and the cauchy-schwarz inequality.
My attempt: Since i wasn't seeing any useful way to apply cauchy-schwarz i squared the l-h-s and used the FTC using that $f$ is radial, like this: $$ ||x|f(x)|^2=\int_0^{|x|}\partial_r(r^2f(r)\overline{f(r)})dr=\int_0^{|x|}2r|f(r)|^2+2\Re \left(r^2\overline{f(r)}\partial_rf(r)\right)dr,$$ and now i could try to apply cauchy schwarz, but i think this isn't going anywhere because there are too many terms i don't want and don't know how to get rid of them. Any tips will be appreciated.