Find The Function

\mathbf{\Rightarrow\int_{0}^{1}f(x).\left\{x-f(x)\right\}dx=\frac{1}{12}}$\\\\ $\mathbf{\Rightarrow\int_{0}^{1}\left\{x.f(x)-(f(x))^2\right\}dx=\int_{0}^{1}\frac{x^2}{4}dx}$\\\\ $\mathbf{\Rightarrow\int_{0}^{1}\left\{f(x)-\frac{x}{2}\right\}^2dx=0}$\\\\ \textbf{So Which is Possible only when $\mathbf{f(x)-\frac{x}{2}}=0$}$\\\\ $\textbf{So $\boxed{\boxed{\mathbf{f(x)=\frac{x}{2}}}}$}

OR by Using C.S Inequality.

Indeed!!