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Multivariable Optimization - Closest point on surface

  1. Jul 7, 2014 #1
    1. The problem statement, all variables and given/known data

    Find the first-octant point P(x,y,z) on the surface closest to the given fixed point Q (0,0,0).
    The surface x2y2z=4

    2. Relevant equations

    gif.latex?d%3D%5Csqrt%7Bx%5E2%2By%5E2%2B(%5Cfrac%7B4%7D%7Bx%5E2y%5E2%7D)%5E2%7D.gif is the distance along PQ.

    EDIT:

    2By%5E2%2B(%5Cfrac%7B4%7D%7Bx%5E2y%5E2%7D)%5E2%3Dx%5E2%2By%5E2%2B%5Cfrac%7B16%7D%7Bx%5E4y%5E4%7D.gif

    3. The attempt at a solution

    gif.latex?z%3D%5Cfrac%7B4%7D%7Bx%5E2y%5E2%7D.gif

    frac%7B64%7D%7Bx%5E5y%5E4%7D%3D0%5C%5C%0A%5C%5Cx%3D%5Cfrac%7B2%7D%7By%5E%5Cfrac%7B4%7D%7B6%7D%7D.gif

    I get stuck here every time. I feel like I'm just selling myself short here, but I don't know how to resolve the situation for when the critical point has a variable in it.
     
    Last edited: Jul 7, 2014
  2. jcsd
  3. Jul 7, 2014 #2

    Mentallic

    User Avatar
    Homework Helper

    Your derivative is incorrect. Try again, more slowly this time.

    When you've done that, then take the partial with respect to y (this should be easy because the problem is symmetric, so you just need to swap the x and y variables in your partial with respect to x) and set that equal to 0 as well. You then have two equations in two unknowns.
     
  4. Jul 7, 2014 #3
    I feel like I'm probably just making myself problems here, but I can't for the life of me get past this still.

    I was minimizing for the distance squared, which gave the two partials. But how can I solve if I have y in terms of x and x in terms of y?!
     
  5. Jul 7, 2014 #4
    Well I feel like a dummy. Got it now, obviously. Sometimes the problem is made so much more difficult in your own mind!
     
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