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Lagrange multipliers

  1. Sep 23, 2008 #1
    1. The problem statement, all variables and given/known data

    Use lagrange multipliers to find the maximum and minimum values of f subject to the given constraint, if such values exist.
    f(x,y) = x+3y, x2+y2≤2

    2. Relevant equations

    grad f = λ grad g

    3. The attempt at a solution

    to find critical points in the interior region,
    The partial derivative of f(x,y) with respect to x is 1.
    The partial derivative of f(x,y) with respect to y is 3.

    g(x,y) = the constraint = x2+y2≤2, to find critical points on the boundary x2+y2=2
    The partial derivative of g(x,y) with respect to x is 2x.
    The partial derivative of g(x,y) with respect to y is 2y.

    And normally I would set:
    1 = λ 2x and 3 = λ 2y
    λ = 1/2x and λ = 3/2y
    so then I would set the equations equal to each other and solve the equation for x and y.

    What I'm wondering though, Is whether there are actually minimum or maximum values since the partial derivatives of f(x,y) are constants?
  2. jcsd
  3. Sep 23, 2008 #2


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    Science Advisor
    Homework Helper

    grad f is nonzero everywhere. So there are no critical points (local maxs or mins) inside the circle x^2+y^2=2. So the max and/or min must be ON the circle. You've found 1/(2x)=3/(2y), substitute that into x^2+y^2=2 and find the possibilities for x and y.
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