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Max/min of multivariate function

  1. Nov 18, 2014 #1
    1. The problem statement, all variables and given/known data
    max/min of
    f(x,y) = x + y
    constraint xy = 16
    3. The attempt at a solution
    With lagrange multipliers I did
    ## \nabla f = (1,1) ##
    ## \nabla g = (y,x) ##
    ## \nabla f = \lambda \nabla g ##
    ## 1 = \lambda y ##
    ## 1 = \lambda x ##
    Since y=0, x=0 aren't a part of xy = 16 I can isolate for lambda

    ## y = x ##
    ## y^2 = 16 ##
    ## y = \pm 4 ##
    ## y = 4, x = 4##
    ## y = -4, x = -4 ##
    ## f(4,4) = 8 ##
    ## f(-4,-4) = -8 ##
    I got these values, but my answer key says that there are no minimums or maximums, can anyone explain why?
     
  2. jcsd
  3. Nov 18, 2014 #2

    LCKurtz

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    The values you have found are relative min/max points as you move along xy=16. But neither are absolute extrema because x+y gets larger and smaller than both then you let either x or y get large positive or negative.
     
  4. Nov 18, 2014 #3
    Yea I understand since as x approaches infinity, y approaches 0, or x approach negative infintiy y approaches 0, so f(x,y) never has a max or min.
     
  5. Nov 19, 2014 #4

    Ray Vickson

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    In the positive quadrant ##x, y \geq 0## your constrained ##f## does have a minimum, but no maximum. In the third quadrant ##x \leq 0, y \leq 0## the constrained function has a maximum, but no minimum. If we throw out the information about quadrants then, of course, it is true that the constrained ##f## had neither a maximum nor a minimum.
     
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