Results 1 to 6 of 6

Math Help - multivariable maximization

  1. #1
    Member
    Joined
    Jan 2011
    Posts
    88

    multivariable maximization

    Hello all, I got another maximization problem I would like to go over.
    Let 0<a<b. Find points (x, y, z) where a<x<y<z<b that maximize

    u=\frac{xyz}{(a+x)(x+y)(y+z)(z+b)}

    First, I said since a>0, and a is the smallest in the domain, let

    v=ln(u)

    and the maximum should be the same.

    Hence,

    v(x, y, z, a, b)=lnx+lny+lnz-ln(a+x)-ln(x+y)-ln(y+z)-ln(z+b)

    where
    v_1(x, y, z, a, b)=\frac{1}{x}-\frac{1}{a+x}-\frac{1}{x+y}

    v_2(x, y, z, a, b)=\frac{1}{y}-\frac{1}{x+y}-\frac{1}{y+z}

    v_3(x, y, z, a, b)=\frac{1}{z}-\frac{1}{y+z}-\frac{1}{z+b}

    v_4(x, y, z, a, b)=\frac{-1}{a+x}

    v_5(x, y, z, a, b)=\frac{-1}{z+b}

    Does this seem reasonable thus far?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: multivariable maximization

    Quote Originally Posted by quantoembryo View Post
    Hello all, I got another maximization problem I would like to go over.
    Let 0<a<b. Find points (x, y, z) where a<x<y<z<b that maximize

    u=\frac{xyz}{(a+x)(x+y)(y+z)(z+b)}

    First, I said since a>0, and a is the smallest in the domain, let

    v=ln(u)

    and the maximum should be the same.

    Hence,

    v(x, y, z, a, b)=lnx+lny+lnz-ln(a+x)-ln(x+y)-ln(y+z)-ln(z+b)

    where
    v_1(x, y, z, a, b)=\frac{1}{x}-\frac{1}{a+x}-\frac{1}{x+y}

    v_2(x, y, z, a, b)=\frac{1}{y}-\frac{1}{x+y}-\frac{1}{y+z}

    v_3(x, y, z, a, b)=\frac{1}{z}-\frac{1}{y+z}-\frac{1}{z+b}

    v_4(x, y, z, a, b)=\frac{-1}{a+x}

    v_5(x, y, z, a, b)=\frac{-1}{z+b}

    Does this seem reasonable thus far?
    a and b are constants, the last two partial derivatives should not appear.

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jan 2011
    Posts
    88

    Re: multivariable maximization

    Okay, given I have done that, is there an easier way to evaluate this that you can see? It doesn't look like a very nice system to solve.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Jan 2011
    Posts
    88

    Re: multivariable maximization

    The equations were not too bad, however, just one question. When I plug the original equation into wolfram it states there is no maxima.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: multivariable maximization

    Quote Originally Posted by quantoembryo View Post
    The equations were not too bad, however, just one question. When I plug the original equation into wolfram it states there is no maxima.
    The extrema are either calculus like or occur on the boundary of the feasible region, but the way your problem is set up the boundary is not part of the feasible region, so there may well be no solution. If the constraints were a \le x \le y \le z \le b and there is no calculus like maximum in the region, there will be a maximum on the boundary.

    CB
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Jan 2011
    Posts
    88

    Re: multivariable maximization

    I managed to get it, thanks for the help.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Multivariable Maximization Problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 29th 2010, 08:25 PM
  2. need help with Maximization
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 26th 2010, 03:46 AM
  3. Maximization
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: February 25th 2010, 06:08 PM
  4. Maximization
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 13th 2008, 04:55 AM
  5. Maximization
    Posted in the Calculus Forum
    Replies: 6
    Last Post: January 19th 2007, 12:39 AM

Search Tags


/mathhelpforum @mathhelpforum