Results 1 to 3 of 3

Math Help - minimum value

  1. #1
    Member
    Joined
    Dec 2010
    Posts
    85
    Thanks
    2

    Red face minimum value

    If x,y\in\mathbb{R} and x^2+xy+y^2 = 1. Then find Minimum value of x^3y+xy^3+4
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor alexmahone's Avatar
    Joined
    Oct 2008
    Posts
    1,074
    Thanks
    7

    Re: minimum value

    Quote Originally Posted by jacks View Post
    If x,y\in\mathbb{R} and x^2+xy+y^2 = 1. Then find Minimum value of x^3y+xy^3+4
    Using the method of Lagrange multipliers,

    3x^2y+y^3=\lambda(2x+y)

    x^3+3xy^2=\lambda(x+2y)

    Adding, we get

    (x+y)^3=3\lambda(x+y)

    x+y=0 or (x+y)^2=3\lambda

    Case 1: x+y=0

    x^2-x^2+x^2=1

    x=\pm 1

    y=\mp 1

    x^3y+xy^3+4=-1-1+4=2

    Case 2: (x+y)^2=3\lambda

    9x^2y+3y^3=(x+y)^2(2x+y)

    =(x^2+y^2+2xy)(2x+y)

    =2x^3+2xy^2+4x^2y+x^2y+y^3+2xy^2

    =2x^3+y^3+5x^2y+4xy^2

    4x^2y-4xy^2-2x^3+2y^3=0

    4xy(x-y)-2(x^3-y^3)=0

    4xy(x-y)-2(x-y)(x^2+xy+y^2)=0

    (x-y)(4xy-2x^2-2xy-2y^2)=0

    (x-y)(2xy-2x^2-2y^2)=0

    (x-y)(x^2+y^2-xy)=0

    x-y=0 or x^2+y^2-xy=0

    Case 2a: x-y=0

    3x^2=1

    x=\pm\frac{1}{\sqrt{3}}

    y=\pm\frac{1}{\sqrt{3}}

    x^3y+xy^3+4=\frac{2}{9}+4=\frac{38}{9}

    Case 2b: x^2+y^2-xy=0

    2(x^2+y^2)=1

    x^2+y^2=\frac{1}{2}

    2xy=1

    (x-y)^2=-\frac{1}{2}

    No solutions.

    --------------------------------------------------

    So, the minimum value of x^3y+xy^3+4 is 2.
    Last edited by alexmahone; January 7th 2012 at 11:46 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: minimum value

    Quote Originally Posted by jacks View Post
    If x,y\in\mathbb{R} and x^2+xy+y^2 = 1. Then find Minimum value of x^3y+xy^3+4
    Switch to polars and substitute r^4 from the constraint into the objective then simplify to give a standard 1D unconstrained problem.

    CB
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: August 21st 2011, 01:12 PM
  2. Pdf of minimum of (x,c)
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: May 13th 2011, 07:59 AM
  3. The Minimum Value of The Sum
    Posted in the Geometry Forum
    Replies: 2
    Last Post: May 2nd 2010, 10:50 AM
  4. Minimum
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: February 24th 2010, 08:58 PM
  5. minimum
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 2nd 2008, 05:49 PM

Search Tags


/mathhelpforum @mathhelpforum