Let f ( x ) = | 1 + sin 2 x cos 2 x sin 2 x sin 2 x 1 + cos 2 x sin 2 x sin 2 x cos 2 x 1 + sin 2 x | , x [ 6 , 3 ] . If and respectively are the maximum and the minimum values of f , then [2023]

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Solution

Q.
Let

$f (x)$ $=$ $| \begin{matrix} 1 + \sin^{2} x & \cos^{2} x & \sin 2 x \\ \sin^{2} x & 1 + \cos^{2} x & \sin 2 x \\ \sin^{2} x & \cos^{2} x & 1 + \sin 2 x \end{matrix} |$ , $x \in [\frac{π}{6}, \frac{π}{3}] .$

If $α$ and $β$ respectively are the maximum and the minimum values of $f$ , then [2023]

1 $β^{2} + 2 \sqrt{α} = \frac{19}{4}$

2 $α^{2} + β^{2} = \frac{9}{2}$

3 $α^{2} - β^{2} = 4 \sqrt{3}$

4 $β^{2} - 2 \sqrt{α} = \frac{19}{4}$

Ans.
(4)

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