Let f ( θ ) = 3 ( ^ 4 ((3π)/(2) - θ) + ^ 4 ( 3π + θ )) - 2(1 - ^ 2...

Given,$f ( \theta ) = 3\left(\sin^{4}\left(\dfrac{3\pi}{2} - \theta\right) + \sin^{4} ( 3\pi + \theta )\right) - 2\left(1 - \sin^{2}2\theta\right)$

and $S = \left\{\theta \in [ 0, \pi ] : {f}^{'} ( \theta ) = - \dfrac{\sqrt{3}}{2}\right\}$

$f ( \theta ) = 3\left(\sin^{4}\left(\dfrac{3\pi}{2} - \theta\right) + \sin^{4} ( 3\pi + \theta )\right) - 2\left(1 - \sin^{2}2\theta\right)$

$⇒ f ( \theta ) = 3\left(\cos^{4}\theta + \sin^{4}\theta\right) - 2\cos^{2}2\theta$

$⇒ f ( \theta ) = 3\left(1 - \dfrac{1}{2}\sin^{2}2\theta\right) - 2\cos^{2}2\theta$

$⇒ f ( \theta ) = 3 - \dfrac{3}{2}\sin^{2}2\theta - 2\cos^{2}\theta$

$⇒ f\left(\theta\right) = \dfrac{3}{2} - \dfrac{1}{2}\cos^{2}2\theta = \dfrac{3}{2} - \dfrac{1}{2}\left(\dfrac{1 + \cos 4\theta}{2}\right)$

$⇒ f ( \theta ) = \dfrac{5}{4} - \dfrac{\cos 4\theta}{4}$

Now on differentiating we get, ${f}^{'} ( \theta ) = \sin 4\theta$

$⇒ {f}^{'} ( \theta ) = \sin 4\theta = - \dfrac{\sqrt{3}}{2}$

$⇒ 4\theta = n\pi + ( - 1{)}^{n}\dfrac{\pi}{3}$

$⇒ \theta = \dfrac{n\pi}{4} + ( - 1{)}^{n}\dfrac{\pi}{12}$

$⇒ \theta = \dfrac{\pi}{12}, \left(\dfrac{\pi}{4} - \dfrac{\pi}{12}\right), \left(\dfrac{\pi}{2} + \dfrac{\pi}{12}\right), \left(\dfrac{3\pi}{4} - \dfrac{\pi}{12}\right)$

$⇒ 4\beta = \dfrac{\pi}{12} + \dfrac{\pi}{6} + \dfrac{7\pi}{12} + \dfrac{8\pi}{12} = \dfrac{3\pi}{2}$

$⇒ \beta = \dfrac{3\pi}{8} ⇒ f ( \beta ) = \dfrac{5}{4} - \dfrac{cos\dfrac{3\pi}{2}}{4} = \dfrac{5}{4}$