Consider the problem of finding the points on the surface [tex]xy+yz+zx=3[/tex] that are closest to the origin.
1) Use the identity [tex](x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx)[/tex] to prove that [tex]x+y+z[/tex] 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 [tex]x,y,z[/tex] and [tex]\lambda[/tex] whose solutions will give the closest points.
3) Find the points on [tex]xy+yz+zx=3[/tex] that are closest to the origin.
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: