Point closest to origin

[wickeddude12]

[attachment=1:3goc0eal]q.PNG[/attachment:3goc0eal]

So I am trying to minimize x^2+y^2+z^2 with the constraint z-1/(xy)=0. Using the the Lagrange multiplier theorem, I have [attachment=0:3goc0eal]eqns.PNG[/attachment:3goc0eal] where lambda is a scalar. Then I have x^2=y^2. I am not sure where to go from here: I need a point and then I need to prove that it is the minimum distance from the origin. However, the curve is not bounded so the extreme value theorem will not work here to prove that it is the minimum distance.

 

[victoriamath]

I get x=y=z=1 by just using partial differentiation of the square of the distance formula from the origin to (x,y,z) with the contraint that XYZ=1.

 

[wickeddude12]

Problem is solved. I get the points (1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1). They must be at a minimum distance by the Lagrange multiplier theorem since they are not maximums as you can find any other arbitrary point that is further away from the origin.