Question

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

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

Answer 1

Answer: (B)

$\frac{\pi }{2}$

$\ln \left(\sin \frac{\pi }{6}\right)$ $=-\ln 2$
And
$\ln \left(\sin \frac{5\pi }{6}\right)$$=\ln \left(\sin \right(\pi -\frac{\pi }{6}\left)\right)$$=-\ln 2$.
Hence Rolle's theorem is applicable.
Now $f^{\prime }\left(c\right)=0$ or $\frac{\cos x}{\sin x}=0$ or $x=\frac{\pi }{2}$.