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

The given determinant is $f\left(x\right)=\left|\begin{array}{ccc}{\sin }^{2}x& 1+{\cos }^{2}x& \cos 2x\\ 1+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& \sin 2x\end{array}\right|,x\in Ris:$

Add the first two columns $f\left(x\right)=\left|\begin{array}{ccc}2& 1+{\cos }^{2}x& \cos 2x\\ 2& {\cos }^{2}x& \cos 2x\\ 1& {\cos }^{2}x& \sin 2x\end{array}\right|$

$R_1\to R_1-R_2$

The above the determinant becomes $f\left(x\right)=\left|\begin{array}{ccc}0& 1& 0\\ 2& {\cos }^{2}x& \cos 2x\\ 1& {\cos }^{2}x& \sin 2x\end{array}\right|=\cos 2x-2\sin 2x$

The maximum value is $\sqrt{1^2+2^2}=\sqrt{5}$