4.5 Variational calculus

Problem: Find the strong extrema of the functional: \[ \begin{array}{l} F(x)=\int_{0}^{1}\left(\dot{x}^{2}-x^{2}\right) d t \rightarrow \text { extr }, x(0)=1, \\ x(1)=\cos 1 . \end{array} \]