Question

The value of $c$ in mean value theorem for the function $f\left(x\right)=\log \sin x$ in the interval $[\frac{\pi }{6},\frac{5\pi }{6}]$ is=

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{2}$
• C. $\frac{2\pi }{3}$
• D. None of these

Answer 1

The correct option is B $\frac{\pi }{2}$

$f\left(x\right)=\log \sin x$

$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{\sin x}\left(\frac{d}{dx}\sin x\right)$

$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{\sin x}\cos x$

$\Rightarrow f^{\prime }\left(x\right)=\cot x$

$\Rightarrow f^{\prime }\left(c\right)=0$

$\Rightarrow \cot c=0$

$\Rightarrow c=\frac{\pi }{2}$