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Math Help - Absolute Extrema in a region (Calc 3)

  1. #1
    Junior Member
    Joined
    Sep 2009
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    Absolute Extrema in a region (Calc 3)

    <br /> <br />
f(x,y)=x^4 - 4x^2y^2 + y^4<br /> <br />
in the region (x,y): -1<=x<=1  -1<=y<=1<br />

    so far i found the critical points to be (0,0) and (1/2,1/2)

    im stuck trying to find points on the four bounded segments
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  2. #2
    Newbie
    Joined
    Nov 2010
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    Just examine these four cases: x=-1, x=1, y=-1, y=1.

    For example,
    x=-1
    f(-1,y)=1-4y^2 + y^4
    df(-1,y)/dy = 4y^3 -8y
    Equate this to 0 and solve for y.
    4y(y^2 - 2)=0
    y=0, y=sqrt(2) [Ignore, since it is outside the region]

    Do these for all cases, and also check the endpoints (-1,1), (-1,-1), (1,1), (1,-1)

    Of all these values, find out which one is the highest and the lowest.
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