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LaGrange Multiplier Problem

  1. Mar 9, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider the intersection of the elliptic paraboloid Z = X2+4Y2 , and the cylinder X2+Y2= 1. Use Lagrange multipliers to find the highest, and lowest points on the curve of intersection.


    2. Relevant equations
    The gradient equations of both functions.


    3. The attempt at a solution

    I have ∇f= <2X, 8Y> and ∇g= <2X, 2Y>. the constraint equation is X2+Y2 = 1.

    I set the equations equal to get:

    2X = (2X)λ 8Y = (2Y)λ

    When I try to solve it always removes the variable. Where do I go from here to solve?
     
  2. jcsd
  3. Mar 9, 2013 #2

    Ray Vickson

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    Look at the first equation ## 2x = 2x \lambda##. Can you cancel the ##2x## on both sides? Why, or why not?
     
  4. Mar 11, 2013 #3
    I think you can, but it would just leave me with λ = 1.
     
  5. Mar 11, 2013 #4

    Ray Vickson

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    OK, so? Work it through to the end.

    BTW: there is another possibility having λ ≠ 1; can you see why?
     
  6. Mar 11, 2013 #5
    I see that λ could also equal 4, but where do I go from plugging in the lambda values? I just end up with 2x=2x, and 8y=2Y, or do I need to use both λ lambda values at the same time?
     
  7. Mar 11, 2013 #6

    Ray Vickson

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    You have three equations, one involving x and λ, one involving y and λ, and the constraint.
     
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