# Critical points

1. Aug 7, 2005

### pbialos

Given $$n\geq 2, n\in \aleph$$ and $$f(x,y)=a*x^n+c*y^n$$ where $$a*c\not=0$$, determine the nature of the critical points. I found the only critical point at (0,0) and when i tried to use the criteria of the determinant of the hessian matrix to determine the kind of critical point it was, it gave me 0, so i can`t say nothing by this criteria.
I dont know what kind of analysis of the function i am supposed to do, so any help would be much appreciated.

Many Thanks, Paul.

2. Aug 7, 2005

### AKG

You can say something of the critical points. First, consider the options ac < 0 and ac > 0. If ac > 0, then you just have polynomials that are just positive multiples of each other. If the polynomial is of odd order, you have nothing (think f(x) = x³), and if it is of even order, then you have a minimum if a > 0 (and hence c > 0) or a maximum if a < 0 (and hence c < 0). Now if ac < 0, then if n is odd, then you again have nothing since this is the same if n is odd and ac > 0, just rotated by 90o. If n is even, then you'll have a saddle point of type (1-1) and unlike minima and maxima, an upside down saddle point is still a saddle point (whereas a maximum becomes a minimum).