Values of c of Rolle's theorem for `f(x)=sin x-sin 2x` on `[0,pi]`

Options

A. `cos^(-1)((1+sqrt(3))/8)`
B. `cos^(-1)((1+sqrt(35))/8)`
C. `cos^(-1)((1-sqrt(38))/8)`
D. does not exist

Answer

A

Explanation

Given `f(x)=sinx-sin2x` on `[0,pi]`
`f'(x)=cos x-2cos 2x=cosx-[2cos^(2)x-1]`
`:.f'(x)=cosx-4cos^(2)x+2`
Consider `f'(c)=0implies4cos^(2)c-cosc-2=0`
`:.cos c=(1+-sqrt(33))/8`
`impliesc=cos(-1)((1+sqrt(3))/8)`