Question

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

• A. $0$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{3}$

Answer 1

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

Rolle's theorem states that if $f\left(x\right)$ be continuous on $[a,b]$, differentiable on $(a,b)$ and $f\left(a\right)=f\left(b\right)$ then there exists some $c$ between $a$ and $b$ such that $f^{{}^{\prime }}\left(c\right)=0$

Given $f\left(x\right)=\log \left(\sin x\right)$ and $[a,b]=[\frac{\pi }{6},\frac{5\pi }{6}]$

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

Therefore, $f{}^{\prime }\left(c\right)=\cot c=0$

$⟹c=\frac{\pi }{2}$