\(
\begin{array}{l}
f(x) = \sin (x)(\sin (x) + 2\cos (x)) \\
f(x) = \sin ^2 (x) + 2\sin (x)\cos (x) \\
f'(x) = 2\sin (x)\cos (x) + 4\cos ^2 (x) - 2 \\
\left. \begin{array}{l}
f'(x) = 4\cos ^2 (x) + 2\sin (x)\cos (x) - 2 \\
f'(x) = 0 \\
\end{array} \right\} \Rightarrow \\
4\cos ^2 (x) + 2\sin (x)\cos (x) - 2 = 0 \\
2\cos (2x) + 2 + \sin (2x) - 2 = 0 \\
2\cos (2x) + \sin (2x) = 0 \\
2 + \tan (2x) = 0 \\
\tan (2x) = - 2 \\
x = \frac{3}{8}\pi - \frac{1}{2}\arctan \left( {\frac{1}{3}} \right) \\
f\left( {\frac{3}{8}\pi - \frac{1}{2}\arctan \left( {\frac{1}{3}} \right)} \right) = \frac{1}{2} + \frac{1}{2}\sqrt 5 = \varphi \\
\end{array}
\)
