Let $$f(x) = \left| {\matrix{ {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \sin 2x} \cr } } \right|,\,x \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of f, then

Question

Let $$f(x) = \left| {\matrix{ {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \sin 2x} \cr } } \right|,\,x \in \left[ {{\pi \over 6},{\pi \over 3}} \right]$.

If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of f, then

${\alpha ^2} - {\beta ^2} = 4\sqrt 3 $

${\beta ^2} - 2\sqrt \alpha = {{19} \over 4}$

${\beta ^2} + 2\sqrt \alpha = {{19} \over 4}$

${\alpha ^2} + {\beta ^2} = {9 \over 2}$

Solution

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