4.15 Variational calculus

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