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Math Help - Find the extreme values of the function subject to the given constraint

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    Find the extreme values of the function subject to the given constraint

    f(x,y)=x^{2}y, x^{2}+2y^{2}=6

    Ok, so I'm trying to do this using Lagrange multipliers

    So, first I need \nabla f=\lambda \nabla g, right?

    Which is:

    2xyi+x^{2}j=\lambda (2xi+4yj)

    So:

    2xy=\lambda 2x and x^{2}=\lambda 4y

    From here, I have no idea what to do. Any help?

    Edit:

    Ok, I THINK I figured it out. Please someone double check my work

    So,

    y=\lambda and x^{2}=4\lambda^{2}

    We plug these into the constraint and get:

    4\lambda^{2}+2\lambda^{2}=6

    =>\lambda=\pm 1

    Then we plug \lambda back into y=\lambda and x=2\lambda

    So we get,  y=\pm 1 and  x=\pm 2

    Then, plugging these numbers into f(x,y)=x^{2}y

    (-2)^{2}(-1)=-4 and (2)^{2}(1)=4

    Is this all correct?
    Last edited by downthesun01; October 20th 2010 at 11:20 PM.
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