Let f (x) = xex. Use the First Derivative Test to determine if each critical point is a minimum, a maximum, or neither.
First we need to find the critical point(s).
The derivative is
f '(x) = xex + ex = ex(x + 1).
This is zero only when x = -1. Since ex is always positive, the sign of f'(x) is the same as the sign of ( x + 1 ). When x < -1, the quantity ( x + 1 ) is negative, so f'(x) is also. When
x > -1 the quantity ( x + 1 ) is positive, so f'(x) is also. We therefore find this numberline:
Therefore f (x) is decreasing until x approaches -1, then increasing again, so at x = -1 there is a minimum.