Question

Find the increased in which $f\left(x\right)=\sin 3x-\cos 3x,0<x<\pi$ is increasing or decreasing.

Answer 1

Consider the problem

we have

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

$\therefore f^{\prime }\left(x\right)=3\cos 3x+3\sin 3x$

$\begin{array}{c}0<x<\pi \\ \Rightarrow 0<3x<3\pi \\ \Rightarrow 0<x<\pi \end{array}$

Thus, Cosine function is positive in the $Ist$ quadrant and negative in the $2nd$ and sin function is positive in both the quadrant.

Therefore, when

$\begin{array}{c}0<x<\frac{\pi }{2}\\ f^{\prime }\left(x\right)>0\end{array}$

Hence, it is increasing when

$\begin{array}{c}\frac{\pi }{2}<x<\pi \\ f^{\prime }\left(x\right)<0\end{array}$

Hence, it is Decreasing

Therefore, $f\left(x\right)$ in increasing on $\left(0,\frac{\pi }{2}\right)$ and decreasing on $\left(\frac{\pi }{2},\pi \right)$