\[\begin{array}{l}
y = {\left( {\sin x - \cos x} \right)^2} + 2\cos 2x + 3\sin x\cos x\\
\Leftrightarrow y = {\sin ^2}x - 2\sin x\cos x + {\cos ^2}x + 2\cos 2x + 3\sin x\cos x\\
\Leftrightarrow y = 1 + \sin x\cos x + 2\cos 2x\\
\Leftrightarrow y = 1 + \frac{1}{2}\sin 2x + 2\cos 2x\\
\Leftrightarrow \sin 2x + 4\cos 2x = 2y - 2.\\
\Rightarrow pt\,\,co\,\,nghiem\,\,\, \Leftrightarrow {a^2} + {b^2} \ge {c^2}\\
\Leftrightarrow 1 + {4^2} \ge {\left( {2y - 2} \right)^2}\\
\Leftrightarrow 4{y^2} - 8y + 4 \le 17\\
\Leftrightarrow 4{y^2} - 8y - 13 \le 0\\
\Leftrightarrow \frac{{2 - \sqrt {17} }}{2} \le y \le \frac{{2 + \sqrt {17} }}{2}\\
\Rightarrow \left\{ \begin{array}{l}
Min\,\,y = \frac{{2 - \sqrt {17} }}{2}\\
Max\,y = \frac{{2 + \sqrt {17} }}{2}
\end{array} \right..
\end{array}\]
