Find the critical points of $g(x,y) = 4x^{3} - 12xy + 3y^{2} - 18y -5.$

Question

I have the function $g(x,y) = 4x^{3} - 12xy + 3y^{2} - 18y -5.$ The only critical points that I have found for this function are $(-1, 1)$, and $(3, 9)$. But my professor insisted that there are more critical points besides these two.

Can anyone help me find them please?

Answer 1

The critical points occur where the gradient of the scalar field is zero. In this case

$\nabla g(x,y)=(12x^2-12y, 6y-12x-18)=\overrightarrow{0}$

If you solve this system of equations you'll find that the only two points are

$(-1, 1)$ and $(3, 9)$

As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.