Answer
$$\eqalign{
& {\text{Inflection point }}\left( {0,0} \right) \cr
& {\text{Concave upward}}:{\text{ }}\left( {0,2\pi } \right) \cr} $$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \tan \left( {\frac{x}{4}} \right),{\text{ }}\left( {0,2\pi } \right) \cr
& {\text{Calculate the second derivative}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {\tan \left( {\frac{x}{4}} \right)} \right] \cr
& f'\left( x \right) = \frac{1}{4}{\sec ^2}\left( {\frac{x}{4}} \right) \cr
& f''\left( x \right) = \frac{d}{{dx}}\left[ {\frac{1}{4}{{\sec }^2}\left( {\frac{x}{4}} \right)} \right] \cr
& f''\left( x \right) = \frac{2}{4}\sec \left( {\frac{x}{4}} \right)\left[ {\sec \left( {\frac{x}{4}} \right)\tan \left( {\frac{x}{4}} \right)} \right]\left( {\frac{1}{4}} \right) \cr
& f''\left( x \right) = \frac{1}{8}{\sec ^2}\left( {\frac{x}{4}} \right)\tan \left( {\frac{x}{4}} \right) \cr
& \frac{1}{8}{\sec ^2}\left( {\frac{x}{4}} \right)\tan \left( {\frac{x}{4}} \right) = 0 \cr
& {\sec ^2}\left( {\frac{x}{4}} \right)\tan \left( {\frac{x}{4}} \right) = 0 \cr
& {\text{On the interval }}\left( {0,2\pi } \right){\text{ }}{\sec ^2}\left( {\frac{x}{4}} \right)\tan \left( {\frac{x}{4}} \right) = - 1{\text{ for }}x = 0 \cr
& {\text{We only have the interval }}\left( {0,2\pi } \right) \cr
& {\text{Making a table of values }}\left( {{\text{See examples on page 188 }}} \right) \cr} $$
\[\boxed{\begin{array}{*{20}{c}}
{{\text{Interval}}}&{\left( {0,2\pi } \right)} \\
{{\text{Test Value}}}&{x = \frac{\pi }{4}} \\
{{\text{Sign of }}f''\left( x \right)}&{ > 0} \\
{{\text{Conclusion}}}&{{\text{Concave upward}}}
\end{array}}\]
$$\eqalign{
& {\text{The inflection point occurs at }}x = 0 \cr
& f\left( 0 \right) = \tan \left( {\frac{0}{4}} \right) = 0 \cr
& f\left( 0 \right) = 0 \cr
& {\text{Inflection point }}\left( {0,0} \right) \cr
& {\text{Concave upward}}:{\text{ }}\left( {0,2\pi } \right) \cr} $$
