Results 1 to 3 of 3

Math Help - Minimum value

  1. #1
    Junior Member
    Joined
    Mar 2010
    Posts
    52

    Minimum value

    x and y be real numbers

    which satisfy x^2+y^2+xy=1

    then find the minimum value of x^3y+y^3x
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,579
    Thanks
    1418
    Looks to me like a "Lagrange multiplier" problem!

    Let f(x,y)= x^2+ xy+ y^2 and g(x,y)= x^3y+ y^3x.

    For any function f, \nabla f= grad f points in the direction of fastest increase so -\nabla f points in the direction of fastest decrease. If there were no constraints, you could find the minimum moving opposite to the way \nabla f was pointing. If you are constrained to surface you cannot necessarily go in that direction but you could still move in the direction of the projection of the vector onto the tangent plane of the surface. The only time you could NOT do that is if there is NO such projection which is the same as saying that \nabla f is parallel to the normal vector to the surface.

    But for a surface given by g(x,y)= constant, the normal vector is simply \nabla g. Saying that one vector is parallel to the other is saying that one is a multiple of the other. The point on g(x,y)= constant that makes f(x,y) a minimum (or a maximum) must satisfy \nabla f= \lambda\nabla g for some constant \lambda, the "Lagrange multiplier".

    Setting the x and y components of that vector equation equal to each other gives two equations with g(x,y)= constant the third equation you need to solve for the three unknowns, x, y and \lambda. Since the value of \lambda is not relevant to the solution, I find that dividing one equation by another, which immediately eliminates \lambda from the equations, is the best way to start.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Mar 2010
    Posts
    52
    could u plz show me, how u found x and y in terms of \lambda

    i initially tried lik that but cudnt found x and y in terms of \lambda
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: August 21st 2011, 12:12 PM
  2. Pdf of minimum of (x,c)
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: May 13th 2011, 06:59 AM
  3. The Minimum Value of The Sum
    Posted in the Geometry Forum
    Replies: 2
    Last Post: May 2nd 2010, 09:50 AM
  4. Minimum
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: February 24th 2010, 07:58 PM
  5. minimum
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 2nd 2008, 04:49 PM

Search Tags


/mathhelpforum @mathhelpforum