Question

If $f\left(x\right)=2si{n}^{3}x-3si{n}^{2}x+12sinx+5\forall x\in (0,\frac{\pi }{2})$, then

• A. f is increasing in $(0,\frac{\pi }{2})$
• B. f is decreasing in $(0,\frac{\pi }{2})$
• C. f is increasing $(0,\frac{\pi }{4})$ and decreasing in $(\frac{\pi }{4},\frac{\pi }{2})$
• D. f is decreasing is $(0,\frac{\pi }{4})$ and increasing in $(\frac{\pi }{4},\frac{\pi }{2})$

Answer 1

The correct option is A f is increasing in $(0,\frac{\pi }{2})$

$f^{\prime }\left(x\right)=6cosx(si{n}^{2}x-sinx+2)>0\forall \in (0,\frac{\pi }{2})$
Thus f(x) is increasing in $(0,\frac{\pi }{2})$