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

Solution:

Clearly, $f\left(x\right)=\log \sin x$ is continuous on $[\pi /6,5\pi /6]$ and differentiable on $(\pi /6,5\pi /6)\text{. }$Therefore, there exists $c\in (\pi /6,5\pi /6)$ such that

$\begin{array}{l}{f}^{\mathrm{\prime }}\left(c\right)=\frac{f\left(\frac{5\pi }{6}\right)-f\left(\frac{\pi }{6}\right)}{\frac{5\pi }{6}-\frac{\pi }{6}}\\ \Rightarrow \cot c=\frac{-{\log }_{e}2+{\log }_{c}2}{\frac{2\pi }{3}}\\ \Rightarrow \cot c=0\Rightarrow c=\frac{\pi }{2}\in (\pi /6,5\pi /6)\end{array}$