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

• To: mathgroup at smc.vnet.net
• Subject: [mg57300] 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 05:12:37 -0400 (EDT)
• References: <200505230620.CAA04045@smc.vnet.net> <EEA0FBBD-9C31-419A-8D0B-7C73AE4DA32E@akikoz.net> <D9EDDB86-15A8-4865-AF0E-9F89803AE6F2@mimuw.edu.pl>
• Sender: owner-wri-mathgroup at wolfram.com

```Andrzej, thanks so much!  I'm still stumped but I did just put together
n there are 2 roots:
a negative one, -(n+1)^2 / (n (n-1))
and the positive one, (n-1)/n.

The prize is officially being augmented by an order of magnitude!

> I still have not got a proof, but thinking about the above proof for
> the case n=2 some conjectures come to my mind which might just
> possible point tothe the way to the general proof. The main idea is
> Daniel Reeves' one: to show that D[f[x,n],x] is not zero on (0,(n-1)/
> 2). If we look at the above argument for n=2  we see that it amounts
> the fact that D[f[x,n],x]==0 has just two rots, one of which is at
> (n-1)/n and the other is negative. One is tempted to conjecture that
> the same is true for any n: the derivative has just two roots, one is
> at (n-1)/n and the other is negative (I have no good guess at this
> time as to where it might be, in the n=2 example it is at -9/2).
>
> At least for one part of the conjecture there is some confirmation:
>
>
> D[f[x,3],x]/.x->2/3//FullSimplify
>
> 0
>
>
> FullSimplify[D[f[x, 4], x] /. x -> (4 - 1)/4]
>
> 0
>
> although again the general n seems to take for ever. There may also
> be a non-computational proof that the equation can only have two
> roots and that one of them must be negative. I will think about it
> more when I have a little more free time but at the moment i view
> these only as long shot conjectures.
>
> Andrzej

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