Answer
$$\eqalign{
& {\text{Relative minumum at: }}\left( {\frac{{11\pi }}{6},\frac{{11\pi }}{6} - \sqrt 3 } \right),\left( {\frac{{23\pi }}{6},\frac{{23\pi }}{6} - \sqrt 3 } \right) \cr
& {\text{Relative maximum at: }}\left( {\frac{{7\pi }}{6},\frac{{7\pi }}{6} + \sqrt 3 } \right),\left( {\frac{{19\pi }}{6},\frac{{19\pi }}{6} + \sqrt 3 } \right) \cr} $$
Work Step by Step
$$\eqalign{
& h\left( x \right) = x - 2\cos x,{\text{ }}\left[ {0,4\pi } \right] \cr
& {\text{Domain: }}\left[ {0,4\pi } \right] \cr
& {\text{*Calculate the first derivative}} \cr
& h'\left( x \right) = \frac{d}{{dx}}\left[ {x - 2\cos x} \right] \cr
& h'\left( x \right) = 1 + 2\sin x \cr
& {\text{Set }}h'\left( x \right) = 0 \cr
& 1 + 2\sin x = 0 \cr
& \sin x = - \frac{1}{2} \cr
& {\text{For the interval }}\left[ {0,4\pi } \right]{\text{ }}\sin x = - \frac{1}{2}{\text{ when:}} \cr
& x = \frac{{7\pi }}{6},\frac{{11\pi }}{6},\frac{{19\pi }}{6},\frac{{23\pi }}{6} \cr
& *{\text{Calculate the second derivative}} \cr
& h''\left( x \right) = \frac{d}{{dx}}\left[ {1 + 2\sin x} \right] \cr
& h''\left( x \right) = 2\cos x \cr
& {\text{Evaluate the second derivative at the critical values}} \cr
& *h''\left( {\frac{{7\pi }}{6}} \right) = 2\cos \left( {\frac{{7\pi }}{6}} \right) = - \sqrt 3 < 0,{\text{ }} \cr
& {\text{Relative maximum at }}\left( {\frac{{7\pi }}{6},f\left( {\frac{{7\pi }}{6}} \right)} \right) = \left( {\frac{{7\pi }}{6},\frac{{7\pi }}{6} + \sqrt 3 } \right) \cr
& *h''\left( {\frac{{11\pi }}{6}} \right) = 2\cos \left( {\frac{{11\pi }}{6}} \right) = \sqrt 3 > 0,{\text{ }} \cr
& {\text{Relative minimum at }}\left( {\frac{{11\pi }}{6},f\left( {\frac{{11\pi }}{6}} \right)} \right) = \left( {\frac{{11\pi }}{6},\frac{{11\pi }}{6} - \sqrt 3 } \right) \cr
& *h''\left( {\frac{{19\pi }}{6}} \right) = 2\cos \left( {\frac{{19\pi }}{6}} \right) = - \sqrt 3 < 0,{\text{ }} \cr
& {\text{Relative maximum at }}\left( {\frac{{19\pi }}{6},f\left( {\frac{{19\pi }}{6}} \right)} \right) = \left( {\frac{{19\pi }}{6},\frac{{19\pi }}{6} + \sqrt 3 } \right) \cr
& *h''\left( {\frac{{23\pi }}{6}} \right) = 2\cos \left( {\frac{{23\pi }}{6}} \right) = \sqrt 3 > 0,{\text{ }} \cr
& {\text{Relative minimum at }}\left( {\frac{{23\pi }}{6},f\left( {\frac{{23\pi }}{6}} \right)} \right) = \left( {\frac{{23\pi }}{6},\frac{{23\pi }}{6} - \sqrt 3 } \right) \cr} $$
