Please answer the following questions about the function

Question

f(x)=e^(-0.5x^2)
Instructions: If you are asked to find x- or y-values, enter either a number, a list of numbers separated by commas, or None if there aren't any solutions. Use interval notation if you are asked to find an interval or union of intervals, and enter { } if the interval is empty.

(a) Find the critical numbers of f, where it is increasing and decreasing, and its local extrema.
Critical numbers x=
Increasing on the interval
Decreasing on the interval
Local maxima x=
Local minima x=

(b) Find where f is concave up, concave down, and has inflection points.
Concave up on the interval=
Concave down on the interval =
Inflection points x=

Find any horizontal and vertical asymptotes of f.
Horizontal asymptotes y=
Vertical asymptotes x=

My answers were:

Critical numbers x= none
Increasing on the interval = didn't have anything
Decreasing on the interval =(-INF,-0.5)U(-0.5,0.5)U(0.5,INF)
Local maxima x= none
Local minima x= none

Concave up on the interval= (-0.5,0)U(0.5,INF)
Concave down on the interval = (-INF,-0.5)U(0,0.5)
Inflection points x= none

Horizontal asymptotes y= none
Vertical asymptotes x=none

Amost none of them is right, only the local minima and the vertical asymtote, please help, I'm not sure how to do this problem.