Question

If hessian determinant is zero, what else can you perform to determine if point P is saddle, local mininmum or local maximum?

Answer 1

The Hessian determinant of a function f(x,y) is defined as H(x,y) = fxx(x,y)fyy(x,y) - fxy(x,y)fyx(x,y).

The Second Partials Test states that if a function f(x,y) has continuous second partials and fx(x0,y0) = 0 and fy(x0,y0) = 0, then

1. H > 0 and fxx(x0,y0) > 0 implies (x0,y0) is a local minimum;

2. H > 0 and fxx(x0,y0) < 0 implies (x0,y0) is a local maximum;

3. H < 0 implies (x0,y0) is a saddle point;

4. H = 0 then the test is inconclusive.

http://www.math.brown.edu/~banchoff/howison/ma35labs-static-latest/20p1.html

know very little about hessian determinants, but hope this helps...?

Answer 2

Yeah I knew that much from previous math courses. But none of my math prof's ever taught me what happens or which procedure to use if hessian is zero.

Answer 3

I Googled it. It sounds like it gets hairy. It used big words. One that I didn't understand and didn't cover in my Calc III class. I found this off a website:

"Basically, if the Hessian determinant is 0, the higher-order terms may be needed to determine the classification of a critical point, but this is not at all trivial to do in general (with or without Maple). If I were trying to do it by hand, I'd hope there were special features of the problem that I could use. If you know that f(x,y) = 0 only on the lines y=x and y=-x, which divide the xy plane into regions, in each of which f(x,y) is either positive or negative. Thus a critical point that only touches regions where f(x,y) is positive would be a local minimum."