On the nature of saddle points

Question

Given the function:

$D(x,y)=f_{xx}f_{yy}-f^2_{xy}$

If $D(a,b)<0$ then this implies $(a,b)$ is a saddle point.

There are three possible ways for $D<0$:

1)If $f_{xx}f_{yy}<0$. This case is represented in the following three images: This is a surface:

We have $f_{xx}|_{(a,b)}>0$, which means that the orange curve- that results from the intersection of a plane parallel to the $xz$ plane and the surface-has positive convexity, therefore $(a,b)$ is minimum along that orange curve.

We have $f_{yy}|_{(a,b)}<0$, which means that the orange curve- that results from the intersection of a plane parallel to the $yz$ plane and the surface-has negative convexity, therefore $(a,b)$ is maximum along that orange curve.

Since along one curve $(a,b)$ is minimum and along another it's maximum, therefore it's a saddle point.

2) If either $f_{xx}|_{(a,b)}$ or $f_{yy}|_{(a,b)}=0$, where $(a,b)$ is a point of inflection, therefore $D=-f^2_{xy}|_{(a,b)}<0$. This case is shown by the following image:

The equation of this surface is: $f(x,y)=x^3-y^2$. The black dot is $(a,b)$. $f_{xx}|_{(a,b)}=0$ where $(a,b)$ is an inflection point along the orange curve, and $f_{yy}|_{(a,b)}<0$ which means the blue curve has negative concavity therefore $(a,b)$ is maximum along that blue curve. Therefore $(a,b)$ is a saddle point.

3)If $f_{xx}$ and $f_{yy}$ have the same signs i.e:$f_{xx}f_{yy}>0$, and $f_{xx}f_{yy}<f^2_{xy}$. I illustrate this case with the following three images:

Point 3) does not make sense to me at all. If $f_{xx}f_{yy}>0$, then this implies the two orange curves -that result from the intersection between the surface on one hand and xz and yz planes(or planes parallel to them), on the other hand- have the same convexity. So according to this reasoning point 3) should imply that $(a,b)$ is an extremum not a saddle point, shouldn't it?

Answer 1

There are two ways I can think to answer this:

First explanation: The greater the magnitude of $f_{xy}$, the more "saddle-like" the critical point in question. The greater the product $f_{xx}f_{yy}$, the more "bowl-like" the critical point is. When all of these quantities are positive, there is a sort of "competition" for the nature of the critical point, and if $f_{xy}^2 > f_{xx}f_{yy}$ then the saddle-like nature "overwhelms" the bowl like nature.

Second explanation: Perhaps it suffices to note that saddles should remain saddles even when we change coordinates. For example, note that $$ f(x,y) = x^2 + 6xy + y^2 $$ Can be rewritten in the form $$ f(x,y) = 2(x+y)^2 - (x-y)^2 $$ From this, we should expect that $f$ has a saddle at zero if and only if $g(x,y) = 2x^2 - y^2$ has one as well.