1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Optimization using Lagrange multipliers

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data[/b]

    f[tex]\left(x,y\right)[/tex] = x^2 +y^2
    g[tex]\left(x,y\right)[/tex] = x^4+y^4 = 2
    Find the maximum and minimum using Lagrange multiplier

    2. Relevant equations



    3. The attempt at a solution

    grad f = 2xi +2yj
    grad g= 4x^3i + 4y^3j

    grad f= λ grad g
    2x=4x^3λ and 2y= 4y^3λ
    2x^2 = 2y^2
    x^2=y^2
    x= [tex]\pm[/tex]y
    x^4+x^4=2
    x=y= [tex]\pm1[/tex]
    max= 1+1=2 @ [tex]\left(1,1\right)[/tex] and [tex]\left(-1,-1\right)[/tex]

    I don't know how to find the min and not sure about the max above
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Mar 22, 2009
  2. jcsd
  3. Mar 22, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Re: optimization

    My recomendation is that you go back and read the problem carefully! What you have written here makes no sense. Usually you use Lagrange multiplier method maximize or minimize a function subject to some constraint. You have two functions with no constraint. Is one of those, either f or g, supposed to be equal to a number?

     
  4. Mar 22, 2009 #3
    Re: optimization

    yes the constraint function is incorrect, i ommitted =2
     
  5. Mar 22, 2009 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Re: optimization

    You have found that y2= x2 and you know that [itex]x^4+ y^4= 2[/itex]. That tells you that [itex]x= \pm 1[/itex] and [itex]x= \pm 1[/itex]. That gives you four possible points: (1, 1), (-1, -1), (1, -1), and (-1, 1). You might want to consider whether there are both maximum and mimimum values.
     
  6. Mar 22, 2009 #5
    Re: optimization

    the points all equal 2(max). i still need to find the min which according to the answer key is sqrt2, I don't know how to get that.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook