Answer
$$\eqalign{
& {\text{Domain}}:\left[ {0,2\pi } \right] \cr
} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = 2x - 4\sin x,{\text{ }}0 \leqslant x \leqslant 2\pi \cr
& {\text{The domain of the function is }}D:\left[ {0,2\pi } \right] \cr
& \cr
& {\text{*Differentiating }} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {2x - 4\sin x} \right] \cr
& f'\left( x \right) = 2 - 4\cos x \cr
& {\text{Set }}f'\left( x \right) = 0 \cr
& 2 - 4\cos x = 0 \cr
& \cos x = \frac{1}{2} \cr
& {\text{On the interval }}\left[ {0,2\pi } \right]{\text{ }}\cos x = \frac{1}{2}{\text{ at: }}x = \frac{\pi }{3},{\text{ }}x = \frac{{5\pi }}{3} \cr
& \cr
& {\text{*Find the second derivative}} \cr
& f''\left( x \right) = \frac{d}{{dx}}\left[ {2 - 4\cos x} \right] \cr
& f''\left( x \right) = 4\sin x \cr
& {\text{Evaluate }}f''\left( x \right){\text{ at the critical points }}x = \frac{\pi }{3},{\text{ }}x = \frac{{5\pi }}{3} \cr
& f''\left( {\frac{\pi }{3}} \right) = 2\sqrt 3 > 0,{\text{ relative minimum}} \cr
& f\left( {\frac{\pi }{3}} \right) = \frac{{2\pi }}{3} - 2\sqrt 3 \to {\text{Relative minimum at }}\left( {\frac{\pi }{3},\frac{{2\pi }}{3} - 2\sqrt 3 } \right) \cr
& f''\left( {\frac{{5\pi }}{3}} \right) = - 2\sqrt 3 < 0,{\text{ relative maximum}} \cr
& f\left( {\frac{{5\pi }}{3}} \right) = \frac{{10\pi }}{3} + 2\sqrt 3 {\text{,Relative maximum at }}\left( {\frac{{5\pi }}{3},\frac{{10\pi }}{3} + 2\sqrt 3 } \right) \cr
& \cr
& {\text{Set }}f''\left( x \right) = 0 \cr
& 4\sin x = 0 \cr
& \sin x = 0 \cr
& {\text{On the interval }}\left[ {0,2\pi } \right]{\text{ }}\sin x = 0{\text{ at: }}x = 0,{\text{ }}x = \pi ,{\text{ }}x = 2\pi \cr
& f\left( 0 \right) = 0 \cr
& f\left( \pi \right) = 2\pi \cr
& f\left( {2\pi } \right) = 4\pi \cr
& {\text{The inflection points are:}} \cr
& \left( {0,0} \right),\left( {\pi ,2\pi } \right),\left( {2\pi ,4\pi } \right) \cr
& \cr
& {\text{*There are no vertical asymptotes because the denominator}} \cr
& {\text{is never 0}}{\text{.}} \cr
& *{\text{There are no horizontal asymptotes}} \cr
& \cr
&} $$
