Question

Find intervals in which the function given by $f\left(x\right)=\sin 3x,x\in [0,\frac{\pi }{2}]$ is decreasing.

Answer 1

$f\left(x\right)=\sin 3x$

$f\left(x\right)=3\cos 3x$

$f\left(x\right)=0$

$\Rightarrow \cos 3x=0$

$\Rightarrow x=\frac{\pi }{6},\frac{\pi }{2}$

Take $(0,\frac{\pi }{6})+(\frac{\pi }{6},\frac{\pi }{2})$

$x\in (0,\frac{\pi }{6})+x\in (\frac{\pi }{6},\frac{\pi }{2})$

$0\le 3x<\frac{\pi }{2}$

$\Rightarrow \cos 3x<0$

$\cos 3x>0$ when $0\le 3x<\frac{\pi }{2}$

$\Rightarrow f\left(x\right)<0$

$\Rightarrow f\left(x\right)>0$

$\Rightarrow f\left(x\right)$ is increasing in $(0,\frac{\pi }{6})$

$\Rightarrow f\left(x\right)$ is decreasing in $(\frac{\pi }{6},\frac{\pi }{2})$