The maximum value of $f\left(x\right) = \left|\begin{array}{ccc}{\sin }^{2}x& 1+{\cos }^{2}x& \cos 2x\\ 1+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& \sin 2x\end{array}\right|, x\in R$ is

$\begin{array}{rcl} {\mathrm{C}}_1& \to & {\mathrm{C}}_1+{\mathrm{C}}_2\\ \mathrm{f}\left(\mathrm{x}\right)& =& \left|\begin{array}{ccc}2 & 1+{\cos }^2\mathrm{x} & \cos 2\mathrm{x}\\ 2& {\cos }^2\mathrm{x}& \cos 2\mathrm{x}\\ 1& {\cos }^2\mathrm{x}& \sin 2\mathrm{x}\end{array}\right|\\ {\mathrm{R}}_1& \to & {\mathrm{R}}_1-{\mathrm{R}}_2\\ & =& \left|\begin{array}{ccc}0& 1& 0\\ 2 & {\cos }^2\mathrm{x} & \cos 2\mathrm{x}\\ 1& {\cos }^2\mathrm{x}& \sin 2\mathrm{x}\end{array}\right|\\ & =& \cos 2\mathrm{x}-2\sin 2\mathrm{x}\\ \max .\mathrm{f}\left(\mathrm{x}\right)& =& \sqrt{5}\end{array}$