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Lagrange Multiplier Problem

  1. Mar 16, 2014 #1
    1. The problem statement, all variables and given/known data

    Find the extrema of f(x, y) = x2−2xy+ 2y2, subject to the
    constraint x2 +y2 = 1.

    2. Relevant equations

    ∇f(x,y) = λg(x,y)

    3. The attempt at a solution

    This is the work I have thus far:

    Letting g(x,y) = x2+y2-1,

    We obtain the following three equations from the Lagrange Multiplier equation:

    2x-2y = 2λx
    4y-2x = 2λy

    I know that x and y cannot both be zero due to the constraint equation.

    If x = 0:

    4y = 2λy

    --> λ = 2

    If y = 0:

    2x = 2λx

    --> λ = 1

    I have no idea what else to do
  2. jcsd
  3. Mar 16, 2014 #2

    Ray Vickson

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    Using the symbol u instead of λ, you have:
    ##(1-u)x = y## and ##(2-u)y = x##, so ##0 = x(u^2 - 3u + 1)##.
  4. Mar 16, 2014 #3


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    At this point, there's really no reason to consider the cases x=0 and y=0. Nothing in the equations obviously suggests that either of those cases have anything to do with solving the system of equations. However, after you combine the equations the way Ray suggests, you find x=0 could be part of a possible solution. You found this implies that ##y \ne 0## and ##\lambda=2##. The first equation, however, then requires that ##y=0##, which is a contradiction. You can therefore eliminate x=0 as a possibility.
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