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Maximum and Minimum : Langrange multiplier problem

  1. Nov 1, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the maximum and minimum of the function f over the closed and bounded set S. Use langrange multiplier method to find the values of the boundary points.

    2. Relevant equations

    f(x,y) = (1+x+y)2

    S = {(x,y) : x2/4 + y2/16 <= 1}


    3. The attempt at a solution


    First, I set their first order partial derivatives to 0 to get the following

    fx(x,y) = 2(1+x+y)=0
    fy(x,y) = 2(1+x+y)=0.

    It's obvious that I'm not going to be able to find a unique value for my critical points with these two equations, thus, I conclude that there are infinitely many critical points in the interior (I also don't understand the intuition behind this conclusion).

    Next, we check the points that satisfy x2/4 + y2/16 = 1, to see if these are potential extremums. I will use the langrange multiplier method.

    1. 2(1+x+y) = [tex]\lambda[/tex]x/2
    2. 2(1+x+y) = [tex]\lambda[/tex]y/8
    3. x2/4 + y2/16 = 1

    Then the textbook says that solving for [tex]\lambda[/tex] will yield y=-x-1 or y=4x. I know how to get y=4x, but where did y=-x-1 come from? How were they able to derive it from these three equations?

    Thank you very much for your help,

    M
     
  2. jcsd
  3. Nov 1, 2010 #2

    MathematicalPhysicist

    User Avatar
    Gold Member

    Your first case is for lambda different than zero, so you can identify the first two equations and then divide by lambda.
    If lambda equals zero, then you get 1+x+y=0, which is the second solution.
     
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