Why being the discriminant $D<0$ guarantees that there is a saddle point in some direction?

Question

I'm studying multi-variable calculus, and stopped at this theorem:

I'm wondering why being the discriminant $D<0$ guarantees that there is a saddle point in some direction ?

score 1 · Answer 1 · answered Mar 09 '16 at 18:37

1

If $f_{xx}$ and $f_{yy}$ have the same sign, then $D$ can still be positive or negative.

As an example, take $f(x,y) = \pm(x^2 + y^2) + 2 \alpha xy$ where $|\alpha| > 1$.

answered Mar 09 '16 at 18:37

Hans Engler

I plotted your example and found that although both second derivative in x and y directions are positive, a saddle point still exists! – Mohamed Mostafa Mar 09 '16 at 18:49
So, the question may be edited to: Why being the discriminant $D<0$ guarantees that there is a saddle point in some direction? – Mohamed Mostafa Mar 09 '16 at 18:51
I found the answer after several pages in the textbook :) thanks – Mohamed Mostafa Mar 09 '16 at 19:40

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