What kind of Critical Points are the ones in the Picture? I am trying to find all local Maxima and Minima of the function $f(x,y):=xy^2 \cdot exp(-x^2-y^2)$.
I found most already, however I found the set $ \{x=x, \ y=0\} $. When calculating the eigenvalues of the Hessian I get one equal to $0$ so no statement is possible with the method. However its clear the points on that line (red line) are critical:

I have two questions:
How are these points called (so far I only know max. min. and saddle points)?
How to show that they are whatever they are called?
 A: You can classify all critical points in one dimension. They are either a minimum, a maximum or a saddle point. But in higher dimension it is not so easy and there are no names for all the cases, which can appear. Therefore we just name a point, where the gradient of the function is zero a critical point and a critical point is an extremum or it isn't. For non-degenerated critical points where the Hessian neither have zero as an eigenvalue nor is zero an accumulation point of the eigenvalues are a classification if the number of positive eigenvalues of the Hessian is finite. In that case we define the Morse index of a critical value as the number of positive eigenvalue wrt their algebraic multiplicity. A critical point with Morse index 0 is therefore a minimum. In $\mathbb{R}^n$ is a critical point with Morse index $n$ a maximum. And you see that there are $n-1$ more different critical points, but they have no name. They are just characterized by their Morse index.
A: note that $$\left(\frac{1}{\sqrt{2}};-1\right)$$ and $$\left(\frac{1}{\sqrt{2}};1\right)$$ are local maxima
and $$\left(-\frac{1}{\sqrt{2}};-1\right)$$ and $$\left(-\frac{1}{\sqrt{2}};1\right)$$ are local minima
