Find the maximum and minimum values of of f(x,y,z) with the following constraints

Question

Find the maximum and minimum values of $f(x,y,z) = x^3 -3y^2 + 4z$ subject to the constraint $g(x,y,z)=x + yz - 4 = 0$.

So far I've plugged this into Mathematica and it returned 6 complex conjugate coordinates and one real $(4.00087, 0.0832872,-0.0104076)$.

I did a bit of manipulation of the constraint and found a second point $(4.00087, -0.0832872, 0.0104076)$ but when I evaluated both the values differed by less than 1.

Does this indicate that the function has no extreme values given the constraint?

Answer 1

The function is unbounded below and above. If you set $x=4$ and $y=0$, any $z$ satisfies the constraints while the objective function is $64+4z$.