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

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