\[y = \sin 2x + \sqrt 3 {\cos ^2}x + 1\]

\[ = \sin 2x + \sqrt 3 \left( {\frac{{1 + \cos 2x}}{2}} \right) + 1\]

\[ = sin2x + \frac{{\sqrt 3 }}{2}\cos 2x + 1 + \frac{{\sqrt 3 }}{2}\]

\[ = \frac{{\sqrt 7 }}{2}\left( {\sin 2x.\frac{{2\sqrt 7 }}{7} + \frac{{\sqrt {21} }}{7}\cos 2x} \right) + 1 + \frac{{\sqrt 3 }}{2}\]

\[ = \frac{{\sqrt 7 }}{2}.\sin \left( {2x + a} \right) + 1 + \frac{{\sqrt 3 }}{2}\]

Với \[\cos a = \frac{{2\sqrt 7 }}{7}\]; \[\sin a = \frac{{\sqrt {21} }}{7}\]

\[ \Rightarrow - \frac{{\sqrt 7 }}{2} + 1 + \frac{{\sqrt 3 }}{2} \le y \le \frac{{\sqrt 7 }}{2} + 1 + \frac{{\sqrt 3 }}{2}\]

 Vậy GTNN của y là \[ - \frac{{\sqrt 7 }}{2} + 1 + \frac{{\sqrt 3 }}{2}\], khi sin(2x + a) = –1

\( \Leftrightarrow 2x + a = - \frac{\pi }{2} + k2\pi \left( {k \in \mathbb{Z}} \right)\)\( \Leftrightarrow x = - \frac{a}{2} - \frac{\pi }{4} + k\pi \left( {k \in \mathbb{Z}} \right)\), với \[\cos a = \frac{{2\sqrt 7 }}{7}\]; \[\sin a = \frac{{\sqrt {21} }}{7}\]

 Vậy GTLN của y là \[\frac{{\sqrt 7 }}{2} + 1 + \frac{{\sqrt 3 }}{2}\], khi sin(2x + a) = 1

\( \Leftrightarrow 2x + a = \frac{\pi }{2} + k2\pi \left( {k \in \mathbb{Z}} \right)\)\( \Leftrightarrow x = - \frac{a}{2} + \frac{\pi }{4} + k\pi \left( {k \in \mathbb{Z}} \right)\), với \[\cos a = \frac{{2\sqrt 7 }}{7}\]; \[\sin a = \frac{{\sqrt {21} }}{7}\]