$\begin{array}{l}
- 1 \le \sin \left( {\frac{x}{2} - \frac{\pi }{4}} \right) \le 1 \Rightarrow - \sqrt 2 \le \sqrt 2 \sin \left( {\frac{x}{2} - \frac{\pi }{4}} \right) \le \sqrt 2 \\
\Rightarrow - \sqrt 2 - 1 \le \sqrt 2 \sin \left( {\frac{x}{2} - \frac{\pi }{4}} \right) - 1 \le \sqrt 2 - 1
\end{array}$
$\begin{array}{l}
\Rightarrow \min y = - \sqrt 2 - 1\,khi\,\\
\sin \left( {\frac{x}{2} - \frac{\pi }{4}} \right) = - 1 \Leftrightarrow \frac{x}{2} - \frac{\pi }{4} = - \frac{\pi }{2} + k2\pi \Leftrightarrow x = - \frac{\pi }{2} + k4\pi \\
\Rightarrow \max y = \sqrt 2 - 1\,khi\,\\
\sin \left( {\frac{x}{2} - \frac{\pi }{4}} \right) = 1 \Leftrightarrow \frac{x}{2} - \frac{\pi }{4} = \frac{\pi }{2} + k2\pi \Leftrightarrow x = \frac{{3\pi }}{2} + k4\pi
\end{array}$
