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MathGroup Archive 2005

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Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg57315] Re: [mg57278] Solve or Reduce on a monstrosity of an expresssion (and a prize!)
  • From: Daniel Reeves <dreeves at umich.edu>
  • Date: Tue, 24 May 2005 05:12:54 -0400 (EDT)
  • References: <200505230620.CAA04045@smc.vnet.net> <acbec1a4050523183617d2f4e3@mail.gmail.com>
  • Sender: owner-wri-mathgroup at wolfram.com

Yes, but I'm only concerned with the case where the 2nd argument, n, is an
integer >= 2.  Then everything's real.

--- \/   FROM Chris Chiasson AT 05.05.23 21:36 (Today)   \/ ---

> Is your function supposed to produce a complex number for f[x,(n -
> 1)/n]/.{x->1,n->3}
>
> ??
>
> On 5/23/05, Daniel Reeves <dreeves at umich.edu> wrote:
> > Mathemahomies,
> >  I have a beast of a function (though continuously differentiable) that I
> > need to prove is strictly decreasing in a certain range (which I *know* it
> > is just from plotting it).  Every combination I can think of of Reduce and
> > Solve and Simplify with assumptions leaves Mathematica spinning its wheels
> > indefinitely.
> >
> > Do you have any ideas for cajoling Mathematica into crunching through
> > this?
> >
> > Here's the function:
> >
> > f[x_,n_] := 9/2/c[x,n]^2*(n+1)b[x,n]^2 (x-d[x,n])(x-x*d[x,n]+d[x,n]+
> >             d[x,n]^2+n (d[x,n]-1) (x+d[x,n]))
> >
> > where
> >
> > a[x_,n_] := 9*(n+1)^2 + Sqrt[3(n+1)^3 (x^2 (n-1) + 27(n+1))];
> >
> > b[x_,n_] := (a[x,n](n-1) x^2)^(1/3);
> >
> > c[x_,n_] := -3^(2/3) x^2 (n^2-1) + 3^(1/3)(x^2(n^2-1) (9 + 9n +
> >             Sqrt[3(n+1) (x^2(n-1) + 27(n+1))]))^(2/3);
> >
> > d[x_,n_] := c[x,n] / (3 b[x,n] (n+1));
> >
> >
> > Show that f[x,n] is strictly decreasing for x in (0,(n-1)/n) for all
> > integers n >= 2.
> >
> > Note that the limit of f[x,n] as x->0 is (n-1)/(2(n+1)) > 0
> > and f[(n-1)/n,n] == 0.  So it would suffice to show that f' has no roots
> > in (0,(n-1)/n).
> >
> >
> > PS: I have a cool prize for information leading to a solution! (whether or
> > not it actually involves Mathematica)
> >
> > --
> > http://ai.eecs.umich.edu/people/dreeves  - -  google://"Daniel Reeves"
> >
> > Sowmya:   Is this guy a mathematician?
> > Terence:  Worse, an economist.  At least mathematicians are honest about
> >           their disdain for the real world.
> >
> >
>
>
>

-- 
http://ai.eecs.umich.edu/people/dreeves  - -  google://"Daniel Reeves"

"Art is the imposing of a pattern on experience, and our aesthetic
enjoyment is recognition of the pattern." -- Alfred North Whitehead


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