Answer
$$\eqalign{
& {\text{Inflection points: }}\left( {\frac{\pi }{2},\frac{\pi }{2}} \right){\text{ and }}\left( {\frac{{3\pi }}{2},\frac{{3\pi }}{2}} \right) \cr
& {\text{Concave downward}}:{\text{ }}\left( {0,\frac{\pi }{2}} \right){\text{ and }}\left( {\frac{{3\pi }}{2},2\pi } \right) \cr
& {\text{Concave upward}}:{\text{ }}\left( {\frac{\pi }{2},\frac{{3\pi }}{2}} \right){\text{ }} \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = x + \cos x,{\text{ }}\left[ {0,2\pi } \right] \cr
& {\text{Calculate the second derivative}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {x + \cos x} \right] \cr
& f'\left( x \right) = 1 - \sin x \cr
& f''\left( x \right) = \frac{d}{{dx}}\left[ {1 - \sin x} \right] \cr
& f''\left( x \right) = - \cos x \cr
& {\text{Set }}f''\left( x \right) = 0 \cr
& - \cos x = 0 \cr
& \cos x = 0 \cr
& {\text{On the interval }}\left[ {0,2\pi } \right]{\text{ }}\cos x = 0{\text{ for }}x = \frac{\pi }{2},{\text{ }}x = \frac{{3\pi }}{2} \cr
& {\text{Set the intervals }}\left( {0,\frac{\pi }{2}} \right),\left( {\frac{\pi }{2},\frac{{3\pi }}{2}} \right),\left( {\frac{{3\pi }}{2},2\pi } \right) \cr
& {\text{Making a table of values }}\left( {{\text{See examples on page 188 }}} \right) \cr} $$
\[\boxed{\begin{array}{*{20}{c}}
{{\text{Interval}}}&{\left( {0,\frac{\pi }{2}} \right)}&{\left( {\frac{\pi }{2},\frac{{3\pi }}{2}} \right)}&{\left( {\frac{{3\pi }}{2},2\pi } \right)} \\
{{\text{Test Value}}}&{x = \frac{\pi }{4}}&{x = \pi }&{x = \frac{{7\pi }}{4}} \\
{{\text{Sign of }}f''\left( x \right)}&{ - \frac{{\sqrt 2 }}{2} < 0}&{1 > 0}&{ - \frac{{\sqrt 2 }}{2} < 0} \\
{{\text{Conclusion}}}&{{\text{C}}{\text{. downward}}}&{{\text{C}}{\text{. upward}}}&{{\text{C}}{\text{. downward}}}
\end{array}}\]
$$\eqalign{
& {\text{The inflection points occur at }}x = \frac{\pi }{2}{\text{ and }}x = \frac{{3\pi }}{2} \cr
& f\left( {\frac{\pi }{2}} \right) = \frac{\pi }{2} + \cos \left( {\frac{\pi }{2}} \right) \to \left( {\frac{\pi }{2},\frac{\pi }{2}} \right) \cr
& f\left( {\frac{{3\pi }}{2}} \right) = \frac{{3\pi }}{2} + \cos \left( {\frac{{3\pi }}{2}} \right) \to \left( {\frac{{3\pi }}{2},\frac{{3\pi }}{2}} \right) \cr
& {\text{Inflection points: }}\left( {\frac{\pi }{2},\frac{\pi }{2}} \right){\text{ and }}\left( {\frac{{3\pi }}{2},\frac{{3\pi }}{2}} \right) \cr
& {\text{Concave downward}}:{\text{ }}\left( {0,\frac{\pi }{2}} \right){\text{ and }}\left( {\frac{{3\pi }}{2},2\pi } \right) \cr
& {\text{Concave upward}}:{\text{ }}\left( {\frac{\pi }{2},\frac{{3\pi }}{2}} \right){\text{ }} \cr} $$
