Maximize distance from the origin

[Rsarette]

Homework Statement

Say that P is a point on the surface xyz=8. Is it true or false that you can always find another point Q on the surface such that Q is further away from the origin than P is?

Homework Equations

∇f(x,y,z)=λ∇g(x,y,z) where ∇g=0

The Attempt at a Solution

Let f(x,y,z)=x²+y²+z²
and g(x,y,z)=xyz-8

Then the ∇f(x,y,z)=λ
2x=λ
2y=λ
2z=λ
Therefore x=y=z
and then plugging into xyz=8
x³=8
x=2

But then the point (2,2,2) is only the square root of 12 away from the origin, where a point like (8,1,1) is the square root of 66 away from the origin. So I think I minimized the distance instead of maximizing the distance fro the origin, but how do I maximize the distance?

 

[Mark44]

Homework Statement

Say that P is a point on the surface xyz=8. Is it true or false that you can always find another point Q on the surface such that Q is further away from the origin than P is?

Homework Equations

∇f(x,y,z)=λ∇g(x,y,z) where ∇g=0

The Attempt at a Solution

Let f(x,y,z)=x²+y²+z²
and g(x,y,z)=xyz-8

Then the ∇f(x,y,z)=λ
2x=λ
2y=λ
2z=λ
Therefore x=y=z
and then plugging into xyz=8
x³=8
x=2

But then the point (2,2,2) is only the square root of 12 away from the origin, where a point like (8,1,1) is the square root of 66 away from the origin. So I think I minimized the distance instead of maximizing the distance fro the origin, but how do I maximize the distance?

What does the graph of xyz = 8 look like? In two dimensions, the graph of xy = 4 is a hyperbola. There is a point on this hyperbola that is closest to the origin, but is it possible to find a point on it that is farthest from the origin? Your problem is similar to this.

Be sure to answer the question that was asked...