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Math Help - Lagrange multipliers to find max and min

  1. #1
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    Lagrange multipliers to find max and min

    The problem statement, all variables and given/known data

    The temperature at a point (x, y) on a metal plate is T(x, y) = 4x^2 − 4xy + y^2 .
    An ant, walking on the plate, traverses a circle of radius 5 centered at the origin.
    Using the method of Lagrange multipliers, find the highest and lowest
    temperatures encountered by the ant.


    The attempt at a solution

    T(x,y) = 4x^2 − 4xy + y^2
    gradient of T = (8x - 4y)i + (2y - 4x)j

    g(x,y) = x^2 + y^2 = 5^2
    gradient of g = (2x)i + (2y)j

    gradient of T = (lambda)gradient of g ----> lambda=#

    8x - 4y = #2x ---->1
    2y - 4x = #2y ---->2

    # = 4-4y = 1-4x

    what am i going to do next?
    Last edited by nameck; November 23rd 2009 at 08:19 PM.
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  2. #2
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    can someone give me and idea to proceed?
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  3. #3
    MHF Contributor Calculus26's Avatar
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    See attachment
    Attached Thumbnails Attached Thumbnails Lagrange multipliers to find max and min-lagrange.jpg  
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  4. #4
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    i get the max is 45 and the min is 5..
    am i calculated it right?
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  5. #5
    MHF Contributor Calculus26's Avatar
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    The max is 125 and the min is 0

    max occur at [-2sqrt(5) ,sqrt(5)] and [2sqrt(5),-sqrt(5)]

    min occur at [sqrt(5),2sqrt(5)] and [-sqrt(5),-2sqrt(5)]
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