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Math Help - Lagrange Multipliers

  1. #1
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    Lagrange Multipliers

    The teacher gave us some problems on the board for us to jot down. He said it would prepare us with the final.

    Problem:

    Optimize the function f(x,y,z)=8x-4z subject to the constraint x^2 + 10y^2 + z^2 = 5

    I am not 100% sure what optimize means and if Lagrange multiplier applies but this is what i have so far.

    f_{x}= 8
    f_{y}= 0
    f_{z}= -4

    g_{x}=2x
    g_{y}= 20y
    g_{z}= 2z

    Combining we have:

    (1)   8 = 2x\lambda
    (2) 0 = 20y\lambda
    (3) -4 = 2z\lambda

    Solving (1) for x:

    (4)  x = \frac{4}{\lambda}
    (5) y = \lambda
    (6) z = \frac{-2}{\lambda}

    Now, add in (4),(5), and (6) for the constraint

    \left(\frac{4}{\lambda}\right)^2 + 10\lambda^2+\left(\frac{-2}{\lambda}\right)^2 = 5

    then

    \frac{16}{\lambda^2} + 10\lambda^2+\frac{4}{\lambda^2} = 5

    So, at this point i am stuck. I don't know how to finish.
    I don't even know if i was suppose to use Lagrange multiplier or if its correct up to that point?

    If someone could let me know if i did the problem correctly and help me finish?
    Thank you so much.
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  2. #2
    MHF Contributor

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    Re: Lagrange Multipliers

    Quote Originally Posted by icelated View Post
    The teacher gave us some problems on the board for us to jot down. He said it would prepare us with the final.

    Problem:

    Optimize the function f(x,y,z)=8x-4z subject to the constraint x^2 + 10y^2 + z^2 = 5

    I am not 100% sure what optimize means and if Lagrange multiplier applies but this is what i have so far.

    f_{x}= 8
    f_{y}= 0
    f_{z}= -4

    g_{x}=2x
    g_{y}= 20y
    g_{z}= 2z

    Combining we have:

    (1)   8 = 2x\lambda
    (2) 0 = 20y\lambda
    (3) -4 = 2z\lambda

    Solving (1) for x:

    (4)  x = \frac{4}{\lambda}
    (5) y = \lambda
    Absolutely not! 0= 20y\lambda gives y= 0.

    (6) z = \frac{-2}{\lambda}

    Now, add in (4),(5), and (6) for the constraint

    \left(\frac{4}{\lambda}\right)^2 + 10\lambda^2+\left(\frac{-2}{\lambda}\right)^2 = 5

    then

    \frac{16}{\lambda^2} + 10\lambda^2+\frac{4}{\lambda^2} = 5

    So, at this point i am stuck. I don't know how to finish.
    I don't even know if i was suppose to use Lagrange multiplier or if its correct up to that point?

    If someone could let me know if i did the problem correctly and help me finish?
    Thank you so much.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2011
    Posts
    117

    Re: Lagrange Multipliers

    oh how did i miss that. So the y dropped out making the problem easier.
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