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

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"$$\left| {\matrix{ { - 2} & { - 2} & 0 \cr 2 & 0 & { - 1} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \cos 2x} \cr } } \right|\left( \matrix{ {R_1} \to {R_1} - {R_2} \hfill \cr \& \,{R_2} \to {R_2} - {R_3} \hfill \cr} \right)$$

= $- 2({\cos ^2}x) + 2(2 + 2\cos 2x + {\sin ^2}x)$

= $4 + 4\cos 2x - 2({\cos ^2}x - {\sin ^2}x)$

$\therefore$ $f(x) = 4 + \underbrace {2\cos 2x}_{\max = 1}$

$\Rightarrow$ $f{(x)_{\max }} = 4 + 2 = 6$"

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

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

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