Answer
$$\eqalign{
& {\bf{a}}.\,\,t = 0 \cr
& {\bf{b}}.\,\,{\text{Does not have}} \cr} $$
Work Step by Step
$$\eqalign{
& {\text{Let }}f\left( t \right) = 100{e^{ - 0.05t}} \cr
& {\bf{a}} \cr
& {\text{Find }}f'\left( t \right) \cr
& f'\left( t \right) = 100\left( { - 0.05{e^{ - 0.05t}}} \right) \cr
& f'\left( t \right) = - 5{e^{ - 0.05t}} \cr
& {\text{Let }}f'\left( t \right) = - 5 \cr
& - 5{e^{ - 0.05t}} = - 5 \cr
& {e^{ - 0.05t}} = 1 \cr
& t = 0 \cr
& \cr
& {\bf{b}}. \cr
& {\text{Let }}f'\left( t \right) = 0 \cr
& - 5{e^{ - 0.05t}} = 0 \cr
& {e^{ - 0.05t}} = 0 \cr
& {\text{There are no values of $t$ at which }}f'\left( t \right) = 0,{\text{ then}} \cr
& {\text{The graph does not have a horizontal tangent line}}{\text{.}} \cr} $$
