Answer
$${\text{No relative extrema}},{\text{ inflection point at }}\left( {0,\frac{1}{2}} \right){\text{.}}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{{{e^x} - {e^{ - x}}}}{2} \cr
& {\text{Calculate the first derivative}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {\frac{{{e^x} - {e^{ - x}}}}{2}} \right] \cr
& f'\left( x \right) = \frac{{{e^x} + {e^{ - x}}}}{2} \cr
& {\text{Set }}f'\left( x \right) = 0 \cr
& {e^x} + {e^{ - x}} = 0 \cr
& {\text{No real solutions, then there are no relative extrema}}{\text{.}} \cr
& \cr
& *{\text{Calculate the second derivative}} \cr
& f''\left( x \right) = \frac{d}{{dx}}\left[ {\frac{{{e^x} + {e^{ - x}}}}{2}} \right] \cr
& f''\left( x \right) = \frac{{{e^x} - {e^{ - x}}}}{2} \cr
& \cr
& {\text{Set }}f''\left( x \right) = 0 \cr
& \frac{{{e^x} - {e^{ - x}}}}{2} = 0 \cr
& x = 0 \cr
& {\text{Inflection point at }}\left( {0,f\left( 0 \right)} \right) \cr
& \left( {0,f\left( 0 \right)} \right) = \frac{{{e^0} - {e^{ - 0}}}}{2} = 0 \cr
& {\text{Inflection point at }}\left( {0,0} \right) \cr
& \cr
& {\text{Graph}} \cr} $$