Results 1 to 3 of 3

Math Help - Lagrange Multipliers

  1. #1
    Junior Member
    Joined
    Dec 2008
    Posts
    70

    Lagrange Multipliers

    Hi, am studying for final exam and has been long time since did lagrange and am stumped on the following,

    Using method of LM find the maximum and minimum values of the function;

    f(x,y,z) = x

    subject to the constraints (x^2) + 2(y^2) + 2(z^2) = 8
    and x + y = z


    I used LM to get the following;

    (1,0,0) = a(2x, 4y, 4z) + b(1, 1, -1)

    Where a = lambda1 and b = lambda 2 (lagrange multipliers)

    but i can't solve the equations to give max/min?

    Any help would be greatly appreciated.
    Cheers.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271
    You have the 3 equations :

    i 1 = 2ax + b

    ii 0 = 4ay + b

    iii 0 = 4az - b

    adding ii and iii we get y = - z using this in the constraint x+y=z we
    obtain:
    x -z = z or x=2z

    Using this in the other constraint x^2 +2y^2 +2z^2 = 8

    We obtain:

    4*z^2 + 2z^2 +2z^2 = 8

    z= +1 , x= + 2 , y = -1,1 (2,-1,1) and (-2,1,-1) are candidates for max and min

    Max is 2 Min is -2 since f(x,y,z) = x
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Calculus26's Avatar
    Joined
    Mar 2009
    From
    Florida
    Posts
    1,271

    Alternative approach

    Alternative approach:

    i 1 = 2ax + b

    ii 0 = 4ay + b

    iii 0 = 4az - b

    adding ii and iii we get y = - z using this in the constraint x+y=z we
    obtain:
    x -z = z or x=2z

    using i and iii we get b= 1/2

    Then:

    x= 1/4a y = -1/8a z= 1/8a

    Using the constraint x^2 +y^2 +z^2 = 8

    a = + 1/8

    a= 1/8 yields (2,-1,1)

    a= -1/8 yields (-2,1,-1)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. lagrange multipliers
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 10th 2009, 03:24 AM
  2. Lagrange Multipliers
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 1st 2009, 12:49 PM
  3. Lagrange Multipliers in 4D
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 23rd 2009, 10:37 AM
  4. Lagrange multipliers
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 22nd 2009, 02:17 PM
  5. Lagrange Multipliers
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 2nd 2009, 10:26 PM

Search Tags


/mathhelpforum @mathhelpforum