The maximum value of f(x) = | cases ^ 2 x & 1 + ^ 2 x & 2x 1 + ^ 2...

f\left(x\right) = \left|\begin{cases}\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{cases}\right|, x \in R

${C}_{1} \to {C}_{1} + {C}_{2}$

= \left|\begin{cases}2 & 1 + \cos^{2}x & \cos 2x \\ 2 & \cos^{2}x & \cos 2x \\ 1 & \cos^{2}x & \sin 2x\end{cases}\right|

${R}_{1} \to {R}_{1} - {R}_{2}$

= \left|\begin{cases}0 & 1 & 0 \\ 2 & \cos^{2}x & \cos 2x \\ 1 & \cos^{2}x & \sin 2x\end{cases}\right|

Expanding w.r.t. ${R}_{1}$

$= - \left(2\sin 2x - \cos 2x\right)$

So, $f\left(x\right) = \cos 2x - 2\sin 2x$

{f\left(x\right)}_\max = \sqrt{1 + 4} = \sqrt{5}