\[\begin{array}{l}
y = \sin 2x + \sqrt 2 - {\sin ^2}2x\\
= - \left( {{{\sin }^2}2x - \sin 2x} \right) + \sqrt 2 \\
= - \left( {{{\sin }^2}2x - 2.\frac{1}{2}\sin 2x + \frac{1}{4}} \right) + \frac{1}{4} + \sqrt 2 \\
= - {\left( {\sin 2x - \frac{1}{2}} \right)^2} + \frac{{4\sqrt 2 + 1}}{4} \le \frac{{4\sqrt 2 + 1}}{4}\\
\Rightarrow Dau\,\, = \,\,\,xay\,\,ra\, \Leftrightarrow \sin 2x = \frac{1}{2}\\
\Leftrightarrow \left[ \begin{array}{l}
2x = \frac{\pi }{6} + k2\pi \\
2x = \frac{{5\pi }}{6} + k2\pi
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{{12}} + \frac{{k\pi }}{4}\\
x = \frac{{5\pi }}{{12}} + \frac{{k\pi }}{4}
\end{array} \right.\,\,\,\left( {k \in Z} \right).\\
Vay\,\,\,Maxy = \frac{{4\sqrt 2 + 1}}{4}.
\end{array}\]
