Last part of extreme value problem

[orange]

Hey hey!

I'm trying to figure out the extreme values of

f(x,y,z) = (x^2 + y^2 - z^2) * e^-(x^2+y^2+z^2)

I do partial differentiation to find where all derivatives are zero, and after some work come to the equation

2(x^2 + y^2 - z^2) = 0

So one obvious extreme point is (0,0,0), but what about x^2 + y^2 - z^2? I figured it's a extreme value circle of radius 1 in the xy-plane, but what about when z isn't 0? Would really appreciate some pointers guys. Thanks! :!)

 

[0rthodontist]

Well, the equation there is the equation of a double cone. You have
x^2 + y^2 = z^2
and for every value of z this determines a circle with radius z.
In the xy plane the circle will have radius 0, not 1.