4.14 Variational calculus

Problem: Investigate the functional to an extremum. \[ \begin{array}{l} I[y(x)]=\int_{0}^{\frac{1}{2}}\left(2 y y^{\prime}+y^{\prime 2}-16 y^{2}\right) d x, \\ y(0)=0, \quad y\left(\frac{1}{2}\right)=0 . \end{array} \]