Let $ ext{f(x)} = \begin{vmatrix} 1 + \sin^2 ext{x} & \cos^2 ext{x} & \sin 2 ext{x} \ \sin^2 ext{x} & 1 + \cos^2 ext{x} & \sin 2 ext{x} \ \sin^2 ext{x} & \cos^2 ext{x} & 1 + \sin 2 ext{x} \end{vmatrix}$, $ ext{x} \in \left[\frac{\pi}{6}, \frac{\pi}{3} ight]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of f, then

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Let $	ext{f(x)} = \begin{vmatrix} 1 + \sin^2 	ext{x} & \cos^2 	ext{x} & \sin 2	ext{x} \ \sin^2 	ext{x} & 1 + \cos^2 	ext{x} & \sin 2	ext{x} \ \sin^2 	ext{x} & \cos^2 	ext{x} & 1 + \sin 2	ext{x} \end{vmatrix}$, $	ext{x} \in \left[\frac{\pi}{6}, \frac{\pi}{3}ight]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of f, then

Accepted Answer

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

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