$\begin{array}{l} y = 2{\sin ^2}x + {\cos ^2}x\\ y = {\sin ^2}x + {\cos ^2}x + {\sin ^2}x\\ y = 1 + {\sin ^2}x\\ - 1 \le \sin x \le 1 \Rightarrow 0 \le {\sin ^2}x \le 1\\ \Rightarrow 1 \le y \le 2\\ \Rightarrow \left\{ \begin{array}{l} \min y = 1 \Rightarrow \sin x = 0 \Rightarrow x = k\pi \\ \max y = 2 \Rightarrow \sin x = 1 \Rightarrow x = \dfrac{\pi }{2} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right) \end{array}$
