Results 1 to 3 of 3

Math Help - Maxima and Minima using Lagrange Multipliers

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    7

    Maxima and Minima using Lagrange Multipliers

    Find the local maxima and minima of the following problem by introducing two Lagrange multipliers:

    f(x_1,x_2,x_3) = x_1 + 2x_2 + 2x_3

    subject to

    x_1^2 + x_2^2 + 4x_3^2 = 2 and x_1^2 + (x_2 - 1)^2 + 4_3^2 = 3
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Jul 2010
    From
    Vancouver
    Posts
    432
    Thanks
    17
    Alright. There are some parts of this problem that should be pretty straightforward. Have you introduced said multipliers? Have you written the equations. It'd be awesome if you can write down the vector equations. (you can use \nabla for [LaTeX ERROR: Convert failed] , unles you know it of course)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2010
    Posts
    7
    Okay, I am stuck. Here is what I have now though....


    Define Constraints:
     g_1(x_1, x_2, x_3) = x_1^2 + x_2^2 + 4x_3^2 = 1 ,  g_2(x_1, x_2, x_3) = x_1^2 + (x_2 - 1)^2 + 4x_3^2 = 3

    Then

     L(x) = x_1 + 2x_2 + 2x_3 + \lambda_1(x_1^2 + x_2^2 + 4x_3^2) + \lambda_2(x_1^2 + (x_2 - 1)^2 + 4x_3^2)

    Critical Points satisfy

     \frac{\partial L}{\partial x_1} = 1 + 2 \lambda_1 x_1 + 2 \lambda_2 x_1 = 0
     \frac{\partial L}{\partial x_2} = 2 + 2 \lambda_1 x_2 + 2 \lambda_2 (2x_2 - 2) = 0
     \frac{\partial L}{\partial x_3} = 2 + 8 \lambda_1 x_3 + 8 \lambda_2 x_3 = 0

    But im having trouble finding the values for x_1, x_2, x_3 and then finding  \lambda_1 , \lambda_2
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Maxima and Minima
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 23rd 2009, 01:03 AM
  2. Maxima and Minima
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 30th 2009, 05:06 AM
  3. Maxima and minima 3
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 28th 2009, 09:30 AM
  4. Maxima/Minima
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 2nd 2008, 08:11 AM
  5. maxima and minima
    Posted in the Calculus Forum
    Replies: 8
    Last Post: November 13th 2008, 05:12 PM

Search Tags


/mathhelpforum @mathhelpforum