Let z=-e-x, then dz/dx=e-x.
y=ez, dy/dx=ezdz/dx=eze-x=(e-x)e^(-e-x).
When dy/dx=0, x→∞ and e-x→0 and e0=1, so dy/dx≈e-x→0.
So there is no value of x which makes dy/dx=0.
When x=5, then z=-e-5~-0.0067 and ez=e-0.0067=0.993~1.
As x increases z→0, so y=ez→1, an asymptote.
The maximum value of y=1 is never attained but is the limit as x→∞.
When x=0, z=-1 and y=e-1=0.37 approx. when x<0, z<-1, y=ez<0.36.
As x→-∞, z→-∞ and ez→0. This shows that 0<y<1.