Select all statements that are correct. There may be more than one correct answer. The statements may appear in what seems to be a random order

A$.$ If the discriminant of $f$ is positive, and $f_{ke}$ is negative, then we have a local maximum.

B$.$ The point $(a,b)$ is a critical point for the multivariable function $f(x,y),$ if both partial derivatives are $0$ at the same time.

C$.$ We cannot have a maximum if the discriminant is zero.

D$.$ If the discriminant is negative at a critical point, then we have a saddle point.

E$.$ The formula for the discriminant of $f(x,y)$ is is $f_{}{f}_{yy}-f_{xy}^2$

F$.$ If a function is a minimum in both the $x$ and $y$ directions, then it is a minimum.

G$.$ If the discriminant of $f$ is positive at a critical point, and $f_{ze}$ is positive, then we have a local minimum.

H$.$ A saddle point has a minimum in one direction and a maximum in a different direction.

I. None of the above