Solving LaGrange Multipliers for Closest Points to Origin on xy+yz+zx=3

[wilcofan3]

Homework Statement

Consider the problem of finding the points on the surface xy+yz+zx=3 that are closest to the origin.

1) Use the identity (x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx) to prove that x+y+z is not equal to 0 for any point on the given surface.

2) Use the method of Lagrange multipliers to find a system of four equations in x,y,z and \lambda whose solutions will give the closest points.

3) Find the points on xy+yz+zx=3 that are closest to the origin.

Homework Equations

The Attempt at a Solution

I'm clueless on what to do for the 1st part (although I imagine it's actually something simple), but I think I have the second part down. Problem is, I think I probably need to use the 1st part for the 3rd somehow.

For the second part, I found the system of four equations to be:

2x=\lambda(y+z)
2y=\lambda(x+z)
2z=\lambda(x+y)
xy+yz+zx=3

 

[Russell Berty]

For part 1: HINT - Use the fact that for any point on the surface xy + yz + xz = 3

For part 3: HINT - Add the left sides of the first three equations and their right sides to make a new equation (then use part 1)

 

[Hurkyl]

Problem is, I think I probably need to use the 1st part for the 3rd somehow.

You can use the first part, even if you don't know why the first part is true.

By the way, part 3 is just a system of equations, solve it the way you normally would. The first part doesn't really affect that -- all ithe first part does is let you make a simplification along the way.

 

[wilcofan3]

You can use the first part, even if you don't know why the first part is true.

By the way, part 3 is just a system of equations, solve it the way you normally would. The first part doesn't really affect that -- all ithe first part does is let you make a simplification along the way.

I feel so stupid, I'm failing at solving this simple system, yet I am pretty sure I know what I'm going to end up with. I'm sure it will be something like x+y=-z that I end up with, because than that would say x+y+z=0 which isn't true, which proves that x=y.

EDIT: Nevermind, it's solved. I don't know why I was blanking on solving the system.