#### Explain solution RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 8 maths

$f(x) \text { is increasing on } (0,\frac{\pi }{2}) \text { and decreasing on }(\frac{\pi }{2},\pi ).$

Given:

$f(x)=log\: sin\: x$

To show:

$\text { We have to show that }f(x) \text { is increasing on } (0,\frac{\pi }{2}) \text { and decreasing on }(\frac{\pi }{2},\pi ).$

Hint:

Use increasing and decreasing property to find increasing and decreasing.

Solution:

Given

$f(x)=log\: sin\: x$

On differentiating both sides w.r.t x we get

\begin{aligned} &\Rightarrow f^{\prime}(x)=\frac{d}{d x}(\log \sin x) \\ &\Rightarrow f^{\prime}(x)=\frac{1}{\sin x} \times \cos x \\ &f^{\prime}(x)=\cot x,\left[\therefore \frac{\cos x}{\sin x}=\cot x\right] \end{aligned}

Taking different region from 0 to $\pi$

\begin{aligned} &\text { Let } x \in\left(0, \frac{\pi}{2}\right) \\ &\Rightarrow \cot x>0 \\ &\Rightarrow f^{\prime}(x)>0 \end{aligned}

$\text { Thus }f(x) \text { increasing }(0,\frac{\pi }{2} ).$

\begin{aligned} &\text { Let } x \in\left( \frac{\pi}{2},\pi \right) \\ &\Rightarrow \cot x< 0 \\ &\Rightarrow f^{\prime}(x)< 0 \end{aligned}

$\text { Thus }f(x) \text { decreasing }(\frac{\pi }{2},\pi ).$

Hence proved.