Question

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

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

Answer 1

Answer: (D)

$f(x)=3 \sin ^{4} x+10 \sin ^{3} x+6 \sin ^{2} x-3, x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$
$f^{\prime}(x)=12 \sin ^{3} x \cos x+30 \sin ^{2} x \cos x+12 \sin x \cos x$
$=6 \sin x \cos x\left(2 \sin ^{2} x+5 \sin x+2\right)$
$=6 \sin x \cos x(2 \sin x+1)(\sin +2)$

decreasing in $\left(-\frac{\pi}{6}, 0\right)$