f '(x) = (-3 - 2x) *e^ (-3x - x²)
2) Set the derivative equal to 0 and find all values of x which satisfy the equation; also find all values of x for which the derivative is undefined.
0 = (-3 - 2x) *e^ (-3x - x²)
Set each factor equal to 0
-3 - 2x = 0
-3 = 2x
-3/2 = x
e^ (-3x - x²) = 0, there is no solution
Our only critical point is x=-3/2
3) Plug in the critical point and the end points into the function. The critical point is -3/2, and the endpoints are -2 and 0, which I got from the interval [-2,0]. This is where you are stuck.
f(0) = e^(-3*0 - 0²) = e^0 = 1
f(-3/2) = e^ (-3 * (-3/2) - (-3/2)²) = e^ (9/2 - 9/4) = e^ (9/4) ≈ 9.488
f(-2) = e^ (-3 * (-2) - (-2)²) = e^ (6 - 4) = e^2 ≈ 7.389
4) From your v-values, pick the least and the greatest y-values and their respective x-values.
Global max:
e^ (9/4) , which occurs at x = -3/2
Global min:
1, which occurs at x=0