Maping from R2 to R3 torus, finding max min and saddle points

[Daveyboy]

Homework Statement

this is from ch9 (functions of several variables)of baby rudin

a,b real b>a>0 define a mapping f=(f₁,f₂),f₃) of R2 into R3 by
f₁(s,t)=(b+acos(s))cos(t)
f₂(s,t)=(b+acos(s))sin(t)
f₂(s,t)=asin(s)
I showed that there are exactly 4 points p in K=image(f) such that

gradf₁(f^-1(p))=0

I am having trouble finding which of these points are local min, max, and saddle points.

The Attempt at a Solution

The points in question correspond to the touple (s,t) when s=(pi)k and t=(pi)j k,j integers.

the points being +/-b +/-a

Okay great, I do not have any developed criteria for the second derivative test, I have no Hessin matrix to work with. I can develop that in my problem, but that seems rather complicated, I have been trying for a while now, and am having trouble.

 

[Daveyboy]

after applying the second derivative test to each component of f I find that at each point of the second derivative of f2 and f3 that the value is zero. So I have zeros at all point of the partial derivatives which is bad because I should have 2 saddle points 1 max and one min. Even more confusing is that in f1 I get two min and two max, no saddle points...
Any ideas