Given that

$\displaystyle F\{y(x)\}= \int\begin{array}{cc}1\\0\end{array} (y'^2-y^2) dx $

with the constraint on y(x) such that

$\displaystyle \int \begin{array}{cc}1\\0\end{array} \sqrt{1+y'^2}dx = \sqrt{2} $

and the end condition y(0)=0 and y(1)=1,

prove that F{y(x)} achieves its minimum value for y = x