Answer
$$c{\text{ values: }} - \frac{{3\pi }}{4}, - \frac{\pi }{4},\frac{\pi }{4},\frac{{7\pi }}{4}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \sin 2x,{\text{ }}\underbrace {\left[ { - \pi ,\pi } \right]}_{\left[ {a,b} \right]} \to a = - \pi ,b = \pi \cr
& {\text{Evaluate }}f\left( x \right){\text{ at the endpoints}} \cr
& f\left( { - \pi } \right) = \sin 2\left( { - \pi } \right) = 0 \cr
& f\left( \pi \right) = \sin 2\left( \pi \right) = 0 \cr
& f\left( { - \pi } \right) = f\left( \pi \right) = 0 \cr
& f{\text{ is continuous on }}\left[ { - \pi ,\pi } \right],{\text{ and }}f{\text{ is differentiable on }} \cr
& \left( { - \pi ,\pi } \right),{\text{ then Rolle's Theorem applies}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {\sin 2x} \right] \cr
& f'\left( x \right) = 2\cos 2x \cr
& 2\cos 2x = 0 \cr
& \cos 2x = 0 \cr
& {\text{For the interval }}\left[ { - \pi ,2\pi } \right]{\text{ }}\cos 2x = 0{\text{ when:}} \cr
& x = - \frac{{3\pi }}{4}, - \frac{\pi }{4},\frac{\pi }{4},\frac{{7\pi }}{4} \cr
& c{\text{ values: }} - \frac{{3\pi }}{4}, - \frac{\pi }{4},\frac{\pi }{4},\frac{{7\pi }}{4} \cr} $$
