If \(f(x)=\log (\sin x), x \in\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]\), then value of 'c'by applyng L.mv.T. is

(B)
Value of ' ' by applying L.M.V.T. is to be found out
\(f(x) =\log (\sin x) \text { on }\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right] \Rightarrow \operatorname{lot} a=\frac{\pi}{6}, b=\frac{5 \pi}{6} \)
\( f^{\prime}(x) =\frac{\cos x}{\sin x}=\cot x \Rightarrow f^{\prime}(c)=\cot c \)
\( f^{\prime}(c) =\frac{f(b)-f(a)}{b-a}=\frac{f\left(\frac{5 \pi}{6}\right)-f\left(\frac{\pi}{6}\right)}{5 \frac{\pi}{6}-\frac{\pi}{6}} \)
\( \cot c =\frac{\log \left[\sin \left(\frac{5 \pi}{6}\right)\right]-\log \left[\sin \left(\frac{\pi}{6}\right)\right]}{\frac{2 \pi}{3}}=\frac{\log \frac{1}{2}-\log \left(\frac{1}{2}\right)}{2 \frac{\pi}{3}} \)
\( \cot c=0 \Rightarrow c=\frac{\pi}{2}\)