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)

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