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Lagrange multipliers

  1. Aug 1, 2010 #1
    1. The problem statement, all variables and given/known data
    Using Lagrange multipliers, find the maximum and minimum values of [tex]f(x,y)=x^3y[/tex] with the constraint [tex]3x^4+y^4=1[/tex].


    2. Relevant equations



    3. The attempt at a solution
    Here is my complete solution. I just wanted to make sure there are no errors and I did it correctly. Thanks for any feedback.

    [tex]\nabla f = \lambda \nabla g[/tex]
    [tex]3x^2y=12\lambda x^3[/tex] and [tex]x^3=4\lambda y^3[/tex]
    Solving these I got [tex]\lambda = \frac{1}{4}[/tex] and [tex]\lambda = -\frac{1}{4}[/tex]

    Putting these values into the equation on the right gives x=y and x=-y. Substituting these into the left equation gives [tex]x=y=\frac{1}{\sqrt{2}}[/tex] and [tex]x=\frac{1}{\sqrt{2}}[/tex], [tex]y = -\frac{1}{\sqrt{2}}[/tex].

    Putting these values into the equation for [tex]f[/tex] gives a maximum of [tex]\frac{1}{4}[/tex] and minimum of [tex]-\frac{1}{4}[/tex].
     
    Last edited: Aug 1, 2010
  2. jcsd
  3. Aug 1, 2010 #2

    arildno

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    Your maximum and minimum values are correct, but you have, for the maxima two soluitons,
    [tex]x=y=\pm\frac{1}{\sqrt{2}}[/tex]
    rather then just one maximum.

    Similarly for the minimum value.
     
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