       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

```

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