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Minimizing a Function

  1. Nov 3, 2008 #1
    1. The problem statement, all variables and given/known data
    Minimize xyz on the unit sphere x2+y2+z2=1


    2. Relevant equations
    Lagrange Method.


    3. The attempt at a solution

    My attempt so far.. Im trying to follow the lagrange method

    I set f(x,y,z)=xyz and g(x,y,z)=x2+y2+z2-1

    g(x,y,z) = 0

    I took the gradient of both functions

    Δf(x,y,z) = (yz)i + (xz)j + (xy)k Δg(x,y,z) = (2x)i + (2y)j + (2z)k

    Then set Δf(x,y,z) = λΔg(x,y,z)

    Giving me the following

    yz = λ2x
    xz = λ2y
    xy = λ2z

    After that i am not sure what to do. Please help.
     
  2. jcsd
  3. Nov 3, 2008 #2

    gabbagabbahey

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    Homework Helper
    Gold Member

    Solve that system of 3 equations for x,y and z in terms of λ. Then use the constraint equation to determine λ and finally plug in your solutions to find the corresponding values of xyz...which ones are minimums? which are maximums?
     
  4. Nov 3, 2008 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Since you are not interested in finding a value for [itex]\lambda[/itex], I recommend dividing one equation by another. For example dividing the first equation by the second gives y/x= x/y or x2= y2 so y= x or y= -x. Similarly, dividing the first equation by the second gives z/x= x/z or x2= z2 so z= x or z= -x. Put those into the condition that x2+ y2+ z2= 1 to determine specific values for x, y, and z.
     
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