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

*To*: mathgroup at smc.vnet.net*Subject*: [mg57278] Solve or Reduce on a monstrosity of an expresssion (and a prize!)*From*: Daniel Reeves <dreeves at umich.edu>*Date*: Mon, 23 May 2005 02:20:47 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

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.

**Follow-Ups**:**Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)***From:*Andrzej Kozlowski <andrzej@akikoz.net>

**Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)***From:*Chris Chiasson <chris.chiasson@gmail.com>