Let $f(\theta) = 3\left[\sin^4\left(\frac{3\pi}{2}-\theta\right) + \sin^4(3\pi+\theta)\right] - 2\left(1-\sin^2 2\theta\right)$ and $S = \left\{\theta \in [0, \pi]: f'(\theta) = -\frac{\sqrt{3}}{2}\right\}$. If $4\beta = \sum_{\theta \in S} \theta$, then $f(\beta)$ is equal to
Step-by-Step Solution
Key Concept: Simplify $f(\theta)$ using trig identities to get $f(\theta) = \frac{5}{4} - \frac{\cos 4\theta}{4}$, differentiate, solve $f'(\theta) = -\sqrt{3}/2$.
$f(\theta) = 5/4 - (\cos 4\theta)/4$. $f'(\theta) = \sin 4\theta = -\sqrt{3}/2$. Solutions in $[0,\pi]$: $\theta = \pi/12, \pi/3-\pi/12=\pi/4, \pi/2+\pi/12, 3\pi/4-\pi/12$. $4\beta = \pi/4+\pi/2+3\pi/4 = 3\pi/2 \Rightarrow \beta = 3\pi/8$. $f(3\pi/8) = 5/4 - \cos(3\pi/2)/4 = 5/4 - 0/4 = 5/4$. Answer: (2)
Correct Answer: $\frac{5}{4}$