Finding the minimum or maximum of a bivariate function when $f_{xx}\times f_{yy}-f_{xy}^2=0$. I see that when $f_{xx}\times f_{yy}-f_{xy}^2<0$ then it is a saddle point.
Also when $f_{xx}\times f_{yy}-f_{xy}^2>0$ then it is a minima or maxima.
What is exactly happening when $f_{xx}\times f_{yy}-f_{xy}^2=0$?

For a regular $f(x,y)$ we have around a point $p_0 = (x_0,y_0)$ with $p = (x,y)$
$$
f(x,y) = f(x_0,y_0) + f_x(p_0)(x-x_0)+f_y(p_0)(y-y_0) + \frac 12(p-p_0)^{\top}J(p_0)(p-p_0)+O(|p-p_0|^3)
$$
If at $p_0$ we have a relative minimum/maximum then $f_x(p_0) = f_y(p_0) = 0$ so the characterization is done with the help of
$$
J(p_0) = \left(\begin{array}{cc}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{array}\right)_{p_0}
$$
if $\det\left(J(p_0)\right) = 0$ then we have that the quadratic associated form is the product of two lines as for example $(x+y)^2$ characterizing a parabolic point. If $J(p_0) = 0$ the qualification should be done with the first non null high order differential included in $O(|p-p_0|^3)$
