stylistic:tone, information:extraction
maximum

If 4 A B − E 2 < 0, {\displaystyle 4AB-E^{2}<0,} the function has no maximum or minimum; its graph forms a hyperbolic paraboloid.

If 4 A B − E 2 > 0, {\displaystyle 4AB-E^{2}>0,} the function has a minimum if both A > 0 and B > 0, and a maximum if both A < 0 and B < 0; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at (x m, y m), {\displaystyle (x_{m},y_{m}),} where:

If 4 A B − E 2 = 0 {\displaystyle 4AB-E^{2}=0} and D E − 2 C B = 2 A D − C E ≠ 0, {\displaystyle DE-2CB=2AD-CE\neq 0,} the function has no maximum or minimum; its graph forms a parabolic cylinder.

If 4 A B − E 2 = 0 {\displaystyle 4AB-E^{2}=0} and D E − 2 C B = 2 A D − C E = 0, {\displaystyle DE-2CB=2AD-CE=0,} the function achieves the maximum/minimum at a line—a minimum if A >0 and a maximum if A <0; its graph forms a parabolic cylinder.