\[\begin{array}{l}
y = {\sin ^4}x + {\cos ^4}x + \sin x\cos x\\
\Leftrightarrow y = 1 - \frac{1}{2}{\sin ^2}2x + \frac{1}{2}\sin 2x\\
\Leftrightarrow y = - \frac{1}{2}{\sin ^2}2x + \frac{1}{2}\sin 2x + 1\\
Dat\,\,t = \sin 2x \Rightarrow t \in \left[ { - 1;\,\,1} \right]\\
\Rightarrow y = - \frac{1}{2}{t^2} + \frac{1}{2}t + 1\\
\Rightarrow y' = - t + \frac{1}{2} = 0\\
\Leftrightarrow t = \frac{1}{2}\\
\Rightarrow \left\{ \begin{array}{l}
y\left( { - 1} \right) = 0\\
y\left( {\frac{1}{2}} \right) = \frac{9}{8}\\
y\left( 1 \right) = 1
\end{array} \right. \Rightarrow \mathop {Min}\limits_{\left[ { - 1;\,\,1} \right]} y = 0\,\,khi\,\,t = - 1 \Leftrightarrow \sin 2x = - 1.
\end{array}\]
