Question

Let $f\left(x\right)=3{\sin }^{4}x+10{\sin }^{3}x+6{\sin }^{2}x-3,$ $x\in [-\frac{\pi }{6},\frac{\pi }{2}]$.Then, $f$ is

• A. decreasing in $(-\frac{\pi }{6},0)$
• B. decreasing in $(0,\frac{\pi }{2})$
• C. increasing in $(-\frac{\pi }{6},0)$
• D. increasing in $(-\frac{\pi }{6},\frac{\pi }{2})$

Answer 1

The correct option is A decreasing in $(-\frac{\pi }{6},0)$

$f\left(x\right)=3{\sin }^{4}x+10{\sin }^{3}x+6{\sin }^{2}x-3$
$\Rightarrow f^{\prime }\left(x\right)=12{\sin }^{3}x\cdot \cos x+30{\sin }^{2}x\cdot \cos x+12\sin x\cdot \cos x$
$=3\sin 2x\cdot (2{\sin }^{2}x+5\sin x+2)$
$=3\sin 2x\cdot (\sin x+2)(2\sin x+1)$
$\therefore$ Critical points are $-\frac{\pi }{6},0$ and $\frac{\pi }{2}$



$\therefore f\left(x\right)$ is decreasing in $(-\frac{\pi }{6},0)$
and increasing in $(0,\frac{\pi }{2})$