Question Number 120089 by ZiYangLee last updated on 29/Oct/20
$$\mathrm{If}\:\mathrm{a}\:\mathrm{continuous}\:\mathrm{function}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{satisfies} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}=\int_{\mathrm{0}} ^{\mathrm{1}} {xf}\left({x}\right){dx}=\mathrm{1} \\ $$$$\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left({f}\left({x}\right)\right)^{\mathrm{2}} {dx}\geqslant\mathrm{4} \\ $$