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

*To*: mathgroup at smc.vnet.net*Subject*: [mg57314] Re: [mg57278] Solve or Reduce on a monstrosity of an expresssion (and a prize!)*From*: Chris Chiasson <chris.chiasson at gmail.com>*Date*: Tue, 24 May 2005 05:12:53 -0400 (EDT)*References*: <200505230620.CAA04045@smc.vnet.net>*Reply-to*: Chris Chiasson <chris.chiasson at gmail.com>*Sender*: owner-wri-mathgroup at wolfram.com

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. > > -- Chris Chiasson http://chrischiasson.com/ 1 (810) 265-3161

**References**:**Solve or Reduce on a monstrosity of an expresssion (and a prize!)***From:*Daniel Reeves <dreeves@umich.edu>