For a function f(x,y), the second derivative is equal to fxx*fyy - (fxy)^2

My book says that if the second derivative is greater than 0, then a max/min exists at that point. If the second derivative is less than 0, then it's a saddle point.

What if the second derivative is equal to 0?

Also, how is it possible to test for the max/min of a function such as f(x,y,z)? I can find the critical points, but I don't know how to test if it's a min or a max.