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
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
Do you have any ideas for cajoling Mathematica into crunching through
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]))
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
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.
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