Extrema of Multi.Varb.Function when Hessian=0

Question

Given is the function $f(x,y)=(y^2-x^2)(y^2-2x^)$ After calculating everything we get the only Critical Point at $P(0,0)$ Plugging it into all the Second Order Partial Derivatives we get the Hessian with only $0$ entries at every Point. So now there are possibilitis to classify Extremas if Determinant $H=0$ e.g using special Lines/Planes to set up more information but is there a way to do the classification with all entries $0$?

Answer 1

There are five critical points, namely $(0,0)$, $(1,\pm\sqrt{3/2})$, and $(2,\pm2)$. The latter four can be classified by means of the Hessian. The origin is a degenerate critical point, but we can analyze it using the factorization $$f(x,y)=(y-x)(y+x)(y^2-2x)\ .\tag{1}$$ The function $f$ is zero along the lines $y=x$ and $y=-x$, as well as on the parabola $y^2-2x=0$. All three of these curves go through the origin, and $f$ changes sign whenever the moving point $(x,y)\ne(0,0)$ crosses one of them. (Example: On one side of the line $y=x$ the factor $(y-x)$ in $(1)$ is $>0$, and on the other side of this line the factor $(y-x)$ is $<0$. All other factors in $(1)$ keep their sign when this line is crossed.) It follows that $f(0,0)=0$, and $f$ assumes both signs in the immediate neighborhood of $(0,0)$. Whether you want to call $(0,0)$ a saddle point is up to you. Some people would call it a monkeys saddle.