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)

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