Let f (x) = xex. If possible, use the second derivative test to determine if each critical point is a maximum, a minimum, or neither.
he first derivative is
f '(x) = xex + ex = ex(x + 1),
so there is one critical point at x = -1. For the second derivative test we need the second derivative, which we can find using the product rule.
At the critical point, we have
Therefore f (x) is concave up at x = -1, so looks like this:
We must have a minimum at x = -1.