$\begin{array}{l} c) - 1 \le \sin x \le 1\\ \Rightarrow 0 \le \sqrt {\sin x} \le 1\\ \Rightarrow \left\{ \begin{array}{l} \min y = 0 \Rightarrow \sin x = 0 \Rightarrow x = k\pi \\ \max y = 1 \Rightarrow \sin x = 1 \Rightarrow x = \dfrac{\pi }{2} + k2\pi \end{array} \right.\\ f){\sin ^4}x - 2{\cos ^2}x + 1\\ = {\sin ^4}x - 2\left( {1 - {{\sin }^2}x} \right) + 1\\ = {\sin ^4}x + 2{\sin ^2}x - 1\\ t = {\sin ^2}x\left( {0 \le t \le 1} \right)\\ \Rightarrow y = {t^2} + 2t - 1\\ \Rightarrow f\left( 0 \right) = - 1,f\left( 1 \right) = 2\\ \Rightarrow \left\{ \begin{array}{l} \min y = - 1 \Rightarrow {\sin ^2}x = 0\\ \max y = 2 \Rightarrow {\sin ^2}x = 1 \end{array} \right.\\ \Rightarrow \left\{ \begin{array}{l} \min y = 0 \Rightarrow x = k\pi \\ \max y = 2 \Rightarrow \sin x = \pm 1 \Rightarrow x = \dfrac{\pi }{2} + k\pi \end{array} \right.\\ g)y = \sin x + \sqrt 3 \cos x + 3\\ y = 2\sin \left( {x + \dfrac{\pi }{3}} \right) + 3\\ - 1 \le \sin \left( {x + \dfrac{\pi }{3}} \right) \le 1\\ \Rightarrow - 2 \le 2\sin \left( {x + \dfrac{\pi }{3}} \right) \le 2\\ \Rightarrow 1 \le y \le 5\\ \Rightarrow \left\{ \begin{array}{l} \max y = 5 \Rightarrow \sin \left( {x + \dfrac{\pi }{3}} \right) = 1\\ \min y = 1 \Rightarrow \sin \left( {x + \dfrac{\pi }{3}} \right) = - 1 \end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} \min y = 1 \Rightarrow x + \dfrac{\pi }{3} = - \dfrac{\pi }{2} + k2\pi \\ \max y = 5 \Rightarrow x + \dfrac{\pi }{3} = \dfrac{\pi }{2} + k2\pi \end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} \min y = 1 \Rightarrow x = - \dfrac{{5\pi }}{6} + k2\pi \\ \max y = 5 \Rightarrow x = \dfrac{\pi }{6} + k2\pi \end{array} \right. \end{array}$
