Question

Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π).

Answer 1

$$\begin{gathered} \mathrm{Here}, \\ f\left(x\right)=\log \sin x \\ \text{Domain of log sin}\mathit{\text{x}}\text{is}\left(0,\mathrm{\pi }\right)\text{.} \\ f\text{'}\left(x\right)=\frac{1}{\sin x}\cos x \\ =\cot x \\ \mathrm{For}x\in \left(0,\frac{\mathrm{\pi }}{2}\right),\mathrm{co}tx>0\left[\because \mathrm{Cot}\mathrm{function}\mathrm{is}\mathrm{positive}\mathrm{in}\mathrm{first}\mathrm{quadrant}\right] \\ \Rightarrow f\text{'}\left(x\right)>0 \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is increasing on}\left(0,\frac{\mathrm{\pi }}{2}\right). \\ \mathrm{For}x\in \left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right),\mathrm{co}tx<0\left[\because \mathrm{Cot}\mathrm{function}\mathrm{is}\mathrm{negative}\mathrm{in}\mathrm{second}\mathrm{quadrant}\right] \\ \Rightarrow f\text{'}\left(x\right)<0 \\ \text{So,}\mathit{\text{f}}\text{(}\mathit{\text{x}})\text{is decreasing on}\left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)\text{.} \end{gathered}$$