Question

The function $f (x)=4 \sin ^{3} x-6 \sin ^{2} x+12 \sin x+100$ is strictly

• A. decreasing in $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$
• B. increasing in $\left(\pi, \frac{3 \pi}{2}\right)$
• C. decreasing in $\left[0, \frac{\pi}{2}\right]$
• D. decreasing in $\left(\frac{\pi}{2}, \pi\right)$

Answer 1

Answer: (D)

$f^{\prime}(x)=\left(12 \sin ^{2} x-12 \sin x+12\right) \cos x$
$f^{\prime}(x)=12\left(\sin ^{2} x-\sin x+1\right) \cos x$
$\sin ^{2} x-\sin x+1>0$
$x \in\left(\frac{\pi}{2}, \pi\right) \cos x < 0$