Answer
$$\eqalign{
& f\left( x \right){\text{ has a relative minimum at }}x = \frac{{4\pi }}{3} \cr
& f\left( x \right){\text{ has a relative maximum at }}x = \frac{{2\pi }}{3}{\text{ }} \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = x + 2\sin x \cr
& \left( {\text{a}} \right){\text{Calculating the first derivative}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {x + 2\sin x} \right] \cr
& f'\left( x \right) = 1 + 2\cos x \cr
& {\text{Calculating the critical points, set }}f'\left( x \right) = 0 \cr
& f'\left( x \right) = \cos 2x \cr
& 1 + 2\cos x = 0 \cr
& \cos x = - \frac{1}{2} \cr
& {\text{On the interval }}\left( {0,2\pi } \right){\text{ }}\cos x = - \frac{1}{2},{\text{ for }}x = \frac{{2\pi }}{3},\frac{{4\pi }}{3} \cr
& \cr
& \left( {\text{b}} \right){\text{Set the intervals: }} \cr
& \left( {0,\frac{{2\pi }}{3}} \right),\left( {\frac{{2\pi }}{3},\frac{{4\pi }}{3}} \right),\left( {\frac{{4\pi }}{3},2\pi } \right) \cr
& {\text{Making a table of values }}\left( {{\text{See examples on page 180 }}} \right) \cr} $$
\[\begin{array}{*{20}{c}}
{{\text{Interval}}}&{{\text{Test Value}}}&{{\text{Sign of }}f'\left( x \right)}&{{\text{Conclusion}}} \\
{\left( {0,\frac{{2\pi }}{3}} \right)}&{x = \frac{\pi }{3}}&{ - \frac{1}{2} < 0}&{{\text{Decreasing}}} \\
{\left( {\frac{{2\pi }}{3},\frac{{4\pi }}{3}} \right)}&{x = \pi }&{1 > 0}&{{\text{Increasing}}} \\
{\left( {\frac{{4\pi }}{3},2\pi } \right)}&{x = \frac{{3\pi }}{2}}&{ - < 0}&{{\text{Decreasing}}}
\end{array}\]
$$\eqalign{
& *f'\left( x \right){\text{ changes from negative to positive at }}x = \frac{{4\pi }}{3},{\text{ so }} \cr
& f\left( x \right){\text{ has a relative minimum at }}x = \frac{{4\pi }}{3}{\text{ }} \cr
& *f'\left( x \right){\text{ changes from positive to negative at }}x = \frac{{2\pi }}{3},{\text{ so }} \cr
& f\left( x \right){\text{ has a relative maximum at }}x = \frac{{2\pi }}{3}{\text{ }} \cr
& \cr
& \left( {\text{c}} \right){\text{Graph}} \cr} $$
