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Question
Show that f(x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π) ?
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Solution
\[\text { Here }, \]
\[f\left( x \right) = \log \sin x\]
\[\text { Domain of log sin x is}\left( 0, \pi \right).\]
\[f'\left( x \right) = \frac{1}{\sin x}\cos x\]
\[ = \cot x\]
\[\text { For x} \in \left( 0, \frac{\pi}{2} \right), \text { cot x} > 0 \left[ \because \text { Cot function is positive in first quadrant }\right]\]
\[ \Rightarrow f'\left( x \right) > 0 \]
\[\text { So,f(x)is increasing on} \left( 0, \frac{\pi}{2} \right) . \]
\[\text { For x }\in \left( \frac{\pi}{2}, \pi \right), \text { cot x }< 0 \left[ \because \text { Cot function is negative in second quadrant } \right]\]
\[ \Rightarrow f'\left( x \right) < 0 \]
\[\text { So,f(x)is decreasing on }\left( \frac{\pi}{2}, \pi \right).\]
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