# Finding critical points

1. May 2, 2007

### erica1451

1. The problem statement, all variables and given/known data
f(x,y,z)=(xy+yz+xz)/(1+x^2+y^2+z^2)
Explain why f has no absolute maximum or minimum. How about critical points?

2. Relevant equations
Hint: it is simplest to make 3 cases: a) x+y+z does not =0 b) x+y+z=0 c) x=y=z=0

3. The attempt at a solution
I did cases b and c, but I'm not sure how to go about doing a. Also, I'm not sure how to explain why the function does not have an absolute max or min.

2. May 5, 2007

### erica1451

Can anyone help?

3. May 5, 2007

### tim_lou

hmmm, simplifying things,

$$f(x,y,z)=\frac{1}{2}\cdot\left[\frac{(x+y+z)^2+1}{1+x^2+y^2+z^2}-1\right]$$
how can you bound f(x,y,z) from below? what about from above? can you make f arbitrarily big?
can you make f(x,y,z) arbitrary close to some values? try some additional cases, suppose x=y=z not equaling zero?