Question

Let $f\left(x\right)=\left|\begin{array}{ccc}{\sin }^{2}x& -2+{\cos }^{2}x& \cos 2x\\ 2+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+\cos 2x\end{array}\right|,x\in [0,\pi ].$Then the maximum value of $f\left(x\right)$ is equal to

Answer 1

Given: $f\left(x\right)=\left|\begin{array}{ccc}{\sin }^{2}x& -2+{\cos }^{2}x& \cos 2x\\ 2+{\sin }^{2}x& {\cos }^{2}x& \cos 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+\cos 2x\end{array}\right|$
$R_2\to R_2-R_1$
$R_3\to R_3-R_1$
$\Rightarrow f\left(x\right)=\left|\begin{array}{ccc}{\sin }^{2}x& -2+{\cos }^{2}x& \cos 2x\\ 2& 2& 0\\ 0& 2& 1\end{array}\right|$
$\Rightarrow f\left(x\right)=2{\sin }^{2}x+4-2{\cos }^{2}x+4\cos 2x$
$\Rightarrow f\left(x\right)=2\cos 2x+4$
$\therefore f(x{)}_{max}=6$