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Math Help - find Extreme values of f(x,y) , subject to g(x,y)

  1. #1
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    find Extreme values of f(x,y) , subject to g(x,y)

    Extreme values of f(x,y) = (x-1)^2 + (y+2)^2
    on the circle x^2 + y^2 = 5

    I thought it was done by fx = λgx => 2(x-1) = (2x)λ

    and fy= λgy => 2(y+2) = (2y)λ

    but λ is meant to be a constant.
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  2. #2
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    Re: find Extreme values of f(x,y) , subject to g(x,y)

    With those two equations and the constraint, you have three equations and three unknowns.
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  3. #3
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    Re: find Extreme values of f(x,y) , subject to g(x,y)

    ah yep.

    so from first one λ = 1 - 1/x
    sub that into the second one gives y = -2x
    sub that into the constraint gives 5x^2 = 5 => x^2 = +1 or -1

    for x=-1
    y= +2 or -2

    and for x= 1
    y= +2 or -2.

    so extreme points at (-1,-2), (-1, 2), (1,-2), (1, 2). Correct?
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  4. #4
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    Re: find Extreme values of f(x,y) , subject to g(x,y)

    No,there are only two points, not four. You have the earlier equation y=-2x.
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  5. #5
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    Re: find Extreme values of f(x,y) , subject to g(x,y)

    oh right, so (-1, 2) and (1,-2)?
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