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Math Help - help Calc 3 problem. Find extrema using legrange multipliers with two? constraints.

  1. #1
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    help Calc 3 problem. Find extrema using legrange multipliers with two? constraints.

    Not sure what I'm doing wrong here.....when I put into wolfram, it said the answer was (1,1,sqrt(2))

    Find the extreme values of
    f(x,y,z)=xyz

    x^2+y^2+z^2=4

    g1=x^2+y^2+z^2-4

    x+y=2

    g2=x+y-2

    so I took the gradient of the function to get:


    grad f=(yz)i+(xz)j+(xy)k

    grad g1=(2x)i+(2y)j+(2z)k

    grad g2=i+j



    grad f=λgrad g1+μgrad g2


    (yz)i+(xz)j+(xy)k=(2xλ+μ)i+(2xλ+μ)j+(2zλ)k
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  2. #2
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    Re: help Calc 3 problem. Find extrema using legrange multipliers with two? constraint

    Quote Originally Posted by theflaktivist View Post
    (yz)i+(xz)j+(xy)k=(2xλ+μ)i+(2xλ+μ)j+(2zλ)k
    You mean,

    (yz)i+(xz)j+(xy)k=(2xλ+μ)i+(2yλ+μ)j+(2zλ)k

    ... but that gives you 3 equations in 5 unknowns, e.g...

    yz = 2xλ+μ

    ... etc. And the g1 and g2 conditions are the other 2 (of the 5 equations in 5 unknowns).

    Push ahead...
    Last edited by tom@ballooncalculus; July 21st 2012 at 06:56 AM.
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  3. #3
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    Re: help Calc 3 problem. Find extrema using legrange multipliers with two? constraint

    Set \nabla f = \lambda \nabla g where g(x,y,z) = x^2 + y^2 + z^2. Alternatively, you can solve it in one step using AM-GM but idk if your calculus professor would approve...
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