$3x^3 +9x^2 -72x+2y^3-12y^2-126y+19$
I want to find critical points and their nature of this equation. When I equated $f_x$ and $f_y$ to 0, I m getting $x=2,x=-4$ for $f_x=0$ and $y=-3,y=7$ for $f_y=0$.
now, when I substitute x=2 in f(x,y) I m getting $2y^3-12y^2-126y=65$ which is a cubic equation. I am stuck here and don't know how to solve this...can somebody help?
https://www.geekandnerd.org/criticalsaddle-point-calculator-for-fxy/ - it shows four critical points by combining above ones.can we always do this?whats the logic behind doing the same?