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Constrained Optimisation

  1. Oct 5, 2012 #1
    1. The problem statement, all variables and given/known data
    The temperature of a point on a unit sphere, centered at the origin, is given by
    T(x,y,y)=xy+yz

    2. Relevant equations
    I know that the equation of a unit sphere is x^2+y^2+x^2=1, which will be the constraint.


    3. The attempt at a solution
    The partial derivatives of T are y, x+z and y respectively.
    Unit circle partial derivatives are 2x, 2y and 2z.

    From a theorem in the lecture notes∇T(x,y,z)=[itex]\lambda[/itex]∇G(x,y,z)
    G being the constraint. With the critical points when these equal 0.

    So I get y-[itex]\lambda[/itex]2x=0
    x+z-[itex]\lambda[/itex]2y=0
    y-[itex]\lambda[/itex]2z=0
    with [itex]\lambda[/itex]2x=[itex]\lambda[/itex]2z

    Firstly, am I on the right track? If so, what is the next move?
    Thanks
     
  2. jcsd
  3. Oct 5, 2012 #2

    gabbagabbahey

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    Gold Member

    Well, you haven't actually stated the question, so I would start with that...
     
  4. Oct 5, 2012 #3

    Ray Vickson

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    The last equality implies either (a) λ = 0; or (b) x = z. Try to see what else happens in both cases (a) and (b).

    RGV
     
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