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Math Help - Max and min values of multivariable functions

  1. #1
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    Max and min values of multivariable functions

    In general I can figure out max and min values, but I'm have some trouble with the added constraints. Here's the problem:

    "Find the maximum and minimum values of the function f(x,y)=x^2+y^2+z^2 subject to the constraint x^4+y^4+z^4=1"
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  2. #2
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    Re: Max and min values of multivariable functions

    Using the Lagrange multiplier method you can form an auxiliary function

    L(x,y,z,\lambda) = (x^2+y^2+z^2)+\lambda (x^4+y^4+z^4-1)

    The stationary points can then be found by solving \nabla L = 0
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  3. #3
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    Re: Max and min values of multivariable functions

    Do i include the partial derivative of gamma in the Gradient to find the points, or do i only use x, y, and z for the gradient still? Sorry we haven't covered much Lagrange yet. Just starting it actually.
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    Re: Max and min values of multivariable functions

    Get them all. Why would we go to the trouble to define an extra variable and then just ignore it?
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    Re: Max and min values of multivariable functions

    I think you mean lambda, yes take the derivative with respect to lambda and x,y,z.

    You only need the result for x,y,z in context of the original function.
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    Re: Max and min values of multivariable functions

    Yeah i meant lambda. I mix up greek letters quite a bit.

    I didn't mean do I ignore it, I've just never seen a gradient include lambda before and I wasn't sure how to include it.
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    Re: Max and min values of multivariable functions

    Once you start to studying the method it will become more clear why its included.

    Good luck!
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    Re: Max and min values of multivariable functions

    Okay so tell me if I get this correct or not. Pardon the lack of notation, I don't know how to type them.

    (Gradient)L=<2x+4(lambda)x^3, 2y+4(lambda)y^3, 2z+4(lambda)z^3, x^4+y^4+z^4-1>

    2x+4(lambda)x^3=0
    2y+4(lambda)y^3=0
    2z+4(lambda)z^3=0
    x^4+y^4+z^4-1=0

    x=sqrt(-1/(2*lambda))
    y=sqrt(-1/(2*lambda))
    z=sqrt(-1/(2*lambda))

    (sqrt(-1/(2*lambda)))^4+(sqrt(-1/(2*lambda)))^4+(sqrt(-1/(2*lambda)))^4-1=0

    lambda=sqrt(3)/2

    And then i plug lambda back into the first three equations that are equal to 0.
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