I have to find and classify the critical points of the function: $$f(x,y) = x^3 +2y^3 - 3x^2 -24y + 6$$
I have said that $$f_x = 3x^2 -6x=0 $$ $$3x(x-2)=0$$ $$x=0, 2$$
$$f_y=6y^2-24=0$$ $$y=±2$$
I don't really know how to link the two x values to the two y values obtained.
A bit of clear-minded critical thinking can go a long way. First of all, I want you to realise that your two-variable function is relatively straightforward since the varibales $x$ and $y$ are not entangled but completely separate.
Now, at any of the four critical points your function will have either a local minimum or a local maximum. So, the ordered pairs (0,2), (0,-2), (2,2), (2,-2) are all critical points of the two-variable function.
Does that make sense or is there still something unclear?
I will let you determine for yourself whether each critical point is a local maximum or a local minimum.