4.14 Cases in discriminant test

By the Lemma, we have the following three cases, which are listed according to the sign of the quadratic form.

(i) Here $\Delta>0$ and $A>0$ so $f(a+h,b+k)\geq f(a,b)$ and we have a local minimum.

(ii) Here $\Delta>0$ and $A<0$ so $f(a+h,b+k)\leq f(a,b)$ and we have a local maximum.

(iii) When $\Delta<0$, the function $f$ increases in some directions and decreases in others from the stationary point, and so the stationary point is a saddle.