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

1) Use the identity $\displaystyle (x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)$ to prove that $\displaystyle x+y+z$ 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 $\displaystyle x,y,z$ and $\displaystyle \lambda$ whose solutions will give the closest points.

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

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:

$\displaystyle 2x=\lambda(y+z)$

$\displaystyle 2y=\lambda(x+z)$

$\displaystyle 2z=\lambda(x+y)$

$\displaystyle xy+yz+zx=3$