Let f(x) = left| begin{array}{ccc} 1 + sin^2 | vedclass

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(C) Given the determinant $f(x) = \left| \begin{array}{ccc} 1 + \sin^2 x & \cos^2 x & 4 \sin 2x \ \sin^2 x & 1 + \cos^2 x & 4 \sin 2x \ \sin^2 x & \

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$6$

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(C) Given the determinant $f(x) = \left| \begin{array}{ccc} 1 + \sin^2 x & \cos^2 x & 4 \sin 2x \ \sin^2 x & 1 + \cos^2 x & 4 \sin 2x \ \sin^2 x & \cos^2 x & 1 + 4 \sin 2x \end{array} ight|$.Applying row operations $R_1 ightarrow R_1 - R_2$ and $R_2 ightarrow R_2 - R_3$:$f(x) = \left| \begin{array}{ccc} 1 & -1 & 0 \ 0 & 1 & -1 \ \sin^2 x & \cos^2 x & 1 + 4 \sin 2x \end{array} ight|$.Expanding along the first row:$f(x) = 1(1(1 + 4 \sin 2x) - (-1)(\cos^2 x)) - (-1)(0(1 + 4 \sin 2x) - (-1)(\sin^2 x)) + 0$$f(x) = 1 + 4 \sin 2x + \cos^2 x + \sin^2 x$Since $\sin^2 x + \cos^2 x = 1$,we get:$f(x) = 1 + 4 \sin 2x + 1 = 2 + 4 \sin 2x$.The maximum value of $\sin 2x$ is $1$.Therefore,the maximum value of $f(x) = 2 + 4(1) = 6$.

Let $f(x) = \left| \begin{array}{ccc} 1 + \sin^2 x & \cos^2 x & 4 \sin 2x \\ \sin^2 x & 1 + \cos^2 x & 4 \sin 2x \\ \sin^2 x & \cos^2 x & 1 + 4 \sin 2x \end{array} \right|$,then the maximum value of $f(x)$ is:

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