I've performed LS fit to data in order to fit the following quadratic function:
$$f(x,y) = A~x^2 + B~y^2 + C~x~y + D~x+E~y +F$$ Now, I would like to check that the fitted function looks like a cup, or half a hose (not an inverse cup, not a saddle). is checking that $A>0$ and $B>0$ is enough?
I don't know what you mean by half a hose. It sounds like you want to know whether the quadratic form $Ax^2+By^2+Cxy$ is positive-definite, which it is if and only if $A\gt0$ and $4AB-C^2\gt0$.
It's not enough to check that $A\gt0$ and $B\gt0$; for any such $A$ and $B$, there is $C$ large enough to make the above determinant negative, which implies one positive and one negative eigenvalue, and thus a saddle point.
