Minimizing the integral

$\begin{aligned} \int_{0}^{1}\bigg(x^2f\big(x\big) - xf\big(x\big)^2\bigg)dx \\ =\int_{0}^{1}\bigg[\frac{x^3}{4} -x\Big(f\big(x\big) - \frac{x}{2}\Big)^2\bigg]dx \\ \end{aligned}$ Maximum value of integral is when f $\big($ x $\big)$ = $\frac{x}{2}$ . Hence maximum value of integral is : $\begin{aligned} \int_{0}^{1}\frac{x^3}{4}dx = \frac{1}{16} = 0.0625 \\ \end{aligned}$