Question

The value of $c$ in Lagrange's theorm for the function $f\left(x\right)=\log \sin x$ in the interval $[\pi /6,5\pi /6]$ is

• A. $\pi /4$
• B. $\pi /2$
• C. $2\pi /3$
• D. none of these

Answer 1

Answer: (B)

$\pi /2$

$f\left(x\right)=logsinx$
$f^{\prime }\left(x\right)=cotx$
By Lagrange's mean value theorem,
$f^{\prime }\left(c\right)=\frac{f\left(\frac{5\pi }{6}\right)-f\left(\frac{\pi }{6}\right)}{\frac{2\pi }{3}}$
$\Rightarrow cotc=\frac{log\left(\frac{1}{2}\right)-log\left(\frac{1}{2}\right)}{\frac{2\pi }{3}}$
$\Rightarrow cotc=0$
$\Rightarrow c=\frac{\pi }{2}$