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Math Help - Calculating extreme values for a multivariable function

  1. #1
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    Calculating extreme values for a multivariable function

    I'm supposed to find the absolute extreme values taken on by f on the set of D.

    f (x, y) = 2x^2+y^2−4x−2y+2, D = {(x, y) : 0 ≤ x ≤ 2,0 ≤ y ≤ 2x}.

    Could someone please teach me how to do this. The book doesn't really do it for me. I need to figure out how to find the extreme values.
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  2. #2
    Super Member fardeen_gen's Avatar
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    f (x, y) = 2x^2 + y^2 − 4x − 2y + 2
    df / dx = 4x -4
    df / dy = 2y -2

    A) set df /dx = 0
    4x -4 = 0
    4x = 4
    x = 1

    B) set df / dy = 0
    2y -2 = 0
    2y = 2
    y = 1
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  3. #3
    MHF Contributor matheagle's Avatar
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    You also need to 'walk' yourself around the boundary, which reduces this to a calculus one problem.
    BUT you should always reduce these...

     2(x^2-2x)+y^2-2y+2 = 2(x^2-2x+1)-2+(y^2-2y+1)-1+2 = 2(x-1)^2+(y-1)^2-1.

    That clearly shows that the GLOBAL min is -1 when x=y=1.
    And your max will occur somewhere on the boundary.
    Last edited by matheagle; April 17th 2009 at 12:10 AM.
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