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Maximize distance from the origin

  1. Mar 14, 2014 #1
    1. The problem statement, all variables and given/known data

    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?

    2. Relevant equations

    ∇f(x,y,z)=λ∇g(x,y,z) where ∇g=0

    3. 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?
     
  2. jcsd
  3. Mar 14, 2014 #2

    Mark44

    Staff: Mentor

    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...
     
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