Here is a function $f(x,y)=x^4 + 6x^2y^2 + y^4 -4x^3 - 12xy^2 + 6x^2 + 6y^2 - 4x + 1$. I've happily proved that $(1,0)$ is a critical point for that function. Now I'd like to decide whether is it a saddle point, a minimum or a maximum. I've seen that some technique that might work is to "move" the origin to this point $(1,0)$. But I don't see how it works.
What I've tried:
I've rewritten the function in term of my new coordinates when moving the point but of course I face the same problem of zero-derivatives. Do I not get the technique? Can you tell me about it?
Thank you!
Perhaps not the method you were expecting, but $$f(x, y) = (x-1)^4+y^4+6y^2(x-1)^2$$ may help you decide...