he first derivative is *f *'(*x*) = *xe*^{x} + *e*^{x} = *e*^{x}(*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. |