Hi there! :)
[tex]\large\boxed{\text{Relative minimum at x = -4}}[/tex]
[tex]g'(x) = (x + 4)e^{x}[/tex]
Find the critical point by setting g'(x) to 0:
[tex]0 = (x + 4)e^{x}[/tex]
Set each factor equal to 0:
[tex]0 = x + 4\\\\-4 = x\\\\0 \neq e^{x}[/tex]
Therefore, the only critical point is at x = -4. Test to see whether this is a relative min or max by plugging in values on both sides into the equation for g'(x):
[tex]g'(-5) = (-5 + 4)e^{-5} = -0.0067, -[/tex]
[tex]g'(-3) = (-3 + 4)e^{-3} = 0.0498, +[/tex]
The graph changes from - to + at x = -4, so there is a relative minimum at x = -4.
