The value of c in Lagrange’s theorem for the function f(x)=logc⁡sin⁡x in the interval [π/6,5π/6], is

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

1. A

$\frac{\pi }{4}$

2. B

$\frac{\pi }{2}$

3. C

$\frac{2\pi }{3}$

4. D

none of these

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