$\begin{array}{l}
y = f\left( x \right) = - 2{\cos ^2}x + 3\cos x - 2\\
y = - 2\left( {{{\cos }^2}x - \dfrac{3}{2}\cos x + 1} \right)\\
y = - 2{\left( {\cos x - \dfrac{3}{4}} \right)^2} - \dfrac{7}{8} \le - \dfrac{7}{8}\\
\Rightarrow \max y = \dfrac{{ - 7}}{8} \Rightarrow \cos x = \dfrac{3}{4}\\
f\left( 1 \right) = - 1,f\left( { - 1} \right) = - 2 + 3\left( { - 1} \right) - 2 = - 7\\
\Rightarrow \min y = - 7 \Rightarrow \cos x = - 1
\end{array}$
