Question

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

Answer 1

$\begin{vmatrix}-2 & -2 & 0 \\ 2 & 0 & -1 \\ \sin ^{2} x & \cos ^{2} x & 1+\cos 2 x\end{vmatrix}$
$\left(R_{1} \rightarrow R_{1}-R_{2} \& R_{2} \rightarrow R_{2}-R_{3}\right)$
$-2\left(\cos ^{2} x\right)+2\left(2+2 \cos 2 x+\sin ^{2} x\right)$
$4+4 \cos 2 x-2\left(\cos ^{2} x-\sin ^{2} x\right)$
$f(x)=4+\underset{max = 1}{2 \cos 2 x}$
$f(x)_{\max }=4+2=6$