I have this multivariable function $f(x,y) = (y-x^2)(y-2x^2)$. Obviously the first step to finding any stationary points would be to find where the derivatives $f_x$ and $fy$ are zero. And this occurs only in the point $P(0,0)$.
Next is to find $f_{xx}$,$f_{yy}$ and $f_{xy}$ and create the Hessian matrix. Okay, I do it and begin to classify $P(0,0)$. It gives inconclusive. Thats fine because there are other ways to classify stationary points.
This is where the main issue begins. My manner of classifying stationary points, is similar to how you might do it for a 2D function. I take the derivative $f_x$ and see what the tendencies are at both sides of $x=0$ knowing $y=0$, I do it similarly for $f_y$. This normally allows me to map the functions along y and x and draw what it might look like as a 3D plot. When i do this for this function, the tendencies around both $x=0$ and $y=0$ look like convex( V shape) parabolas. This makes it look like a minima. However apparently it isnt. Its a weird type of saddle point that looks like this:
After seeing it as a saddle point i tried to look at the tendencies of both point values but now in the derivatives of the other, so, check the tendencies of $f_x$ for $y=0$ when $x=0$, which didnt help, but the opposite one did. The tendency around $x=0$ for $f_y$ when $y=0$ shows it being negative on both sides which means it couldnt be a minima because the function still decreases along y from y's perspective, or at least thats how i understood it.
I want to know if my method for classifying inconclusive stationary points is general enough to allow classification of all problems and, if so, why couldnt i get anything for the tendency of $f_x$ around $y=0$?
If it isnt though, then is there a more general method?
