$\text{ Let }f\left(x\right)\text{ be a function, satisfying }f:[0,1]\to \mathrm{ℝ},\text{ and }$$5{\int }_0^1\left(f\right(x){)}^{2}dx+1=10{\int }_0^1{x}^{2}f(x)dx\text{ then which is/are correct }$

$\begin{array}{l}\text{Given 5}\underset{0}{\overset{1}{\int }}{\left[f\left(x\right)\right]}^{2}dx+1=10{\int }_0^1{x}^{2}f\left(x\right)dx\\ \text{ 5}\underset{0}{\overset{1}{\int }}{\left[f\left(x\right)\right]}^{2}dx+\text{5}\underset{0}{\overset{1}{\int }}{x}^{4}dx=10{\int }_0^1{x}^{2}f\left(x\right)dx\\ \text{5}\underset{0}{\overset{1}{\int }}{\left[f\left(x\right)\right]}^2-2x^{2}f\left(x\right)+{\left(x^2\right)}^{2}dx=0\\ \text{5}\underset{0}{\overset{1}{\int }}{\left[f\left(x\right)-x^2\right]}^{2}dx=0\\ f\left(x\right)=x^2\\ f\left(\frac{1}{2}\right)=\frac{1}{4}\\ \text{The function has maximum value }1 \text{in the given interval}\end{array}$