4.6 Variational calculus

Problem: Find the strong extrema of the functional: \[ \begin{array}{l} F(x)=\int_{4}^{-2}\left(\dot{x}^{2}+4 x^{2}\right) d t \rightarrow \text { extr }, \\ x(-4)=16, \quad x(-2)=4 . \end{array} \]