# LaGrange Multipliers

## 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.

## 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:

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

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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
Staff Emeritus
Gold Member
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. :tongue:

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.

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

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.

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