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

• To: mathgroup at smc.vnet.net
• Subject: [mg57364] Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)
• From: Daniel Reeves <dreeves at umich.edu>
• Date: Wed, 25 May 2005 06:03:14 -0400 (EDT)
• References: <200505230620.CAA04045@smc.vnet.net> <EEA0FBBD-9C31-419A-8D0B-7C73AE4DA32E@akikoz.net> <Pine.LNX.4.58.0505231817420.9452@boston.eecs.umich.edu> <458D701E-37FA-425F-89C4-52A5628E22CF@akikoz.net>
• Sender: owner-wri-mathgroup at wolfram.com

```This is strange:  FindInstance proves that f' has no roots in (0,(n-1)/n)
for all n up to 1024 except for n=685 which seems to choke it.
Multiples of 685 are fine.

I wonder what's special about 685.  There's no obvious difference in the
function -- the conjecture is still true.

Small prize for the n=685 case; big prize for the general n case.

Daniel

PS, a restatement of the problem, thanks to Andrzej:

Show for all integers n >= 2 that the function below, f'[x,n], has no
roots in (0,(n-1)/n). Alternatively, show that its only 2 roots are
-(n+1)^2 / (n (n-1))  and  (n-1)/n.

Here's the function (the derivative of the original function I posted).
Medium prize for finding a simpler form for this.

(3*(-1 + n)*(1 + n)*x*((Sqrt[3]*(-1 + n)*(1 + n)^3*x^2)/
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)] +
2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))*
(x - (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))*
(x + (3^(2/3)*(-1 + n^2)*x^2 - 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))^2/(9*(1 + n)^2*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3)) + (-(3^(2/3)*(-1 + n^2)*x^2) +
3^(1/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*
Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)]))^(2/3))/
(3*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)) -
(x*(-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3)))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)) +
n*(-1 + (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))*
(x + (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))))/
(((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)*
(3^(2/3)*(-1 + n^2)*x^2 - 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))^2) -
(9*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 +
Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(2/3)*
(-2*3^(2/3)*(-1 + n^2)*x +
(2*((Sqrt[3]*(-1 + n^2)^2*x^3)/Sqrt[27 + 54*n - x^2 + n^2*(27 + x^2)] +
2*(-1 + n^2)*x*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)])))/(3^(2/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))*
(x - (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))*
(x + (3^(2/3)*(-1 + n^2)*x^2 - 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))^2/(9*(1 + n)^2*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3)) + (-(3^(2/3)*(-1 + n^2)*x^2) +
3^(1/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*
Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)]))^(2/3))/
(3*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)) -
(x*(-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3)))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)) +
n*(-1 + (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))*
(x + (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))))/
(-(3^(2/3)*(-1 + n^2)*x^2) +
3^(1/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*
Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)]))^(2/3))^3 +
(9*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 +
Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(2/3)*
(1 + ((-1 + n)*x*(-(3^(2/3)*(-1 + n^2)*x^2) +
3^(1/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))*((Sqrt[3]*(-1 + n)*(1 + n)^3*x^2)/
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)] +
2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)])))/(9*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(4/3)) - (-2*3^(2/3)*(-1 + n^2)*x +
(2*((Sqrt[3]*(-1 + n^2)^2*x^3)/Sqrt[27 + 54*n - x^2 +
n^2*(27 + x^2)] + 2*(-1 + n^2)*x*(9 + 9*n +
Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)])))/
(3^(2/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*
Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)))/
(3*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)))*
(x + (3^(2/3)*(-1 + n^2)*x^2 - 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))^2/(9*(1 + n)^2*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3)) + (-(3^(2/3)*(-1 + n^2)*x^2) +
3^(1/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*
Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)]))^(2/3))/
(3*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)) -
(x*(-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3)))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)) +
n*(-1 + (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))*
(x + (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))))/
(2*(3^(2/3)*(-1 + n^2)*x^2 - 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))^2) +
(9*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 +
Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(2/3)*
(x - (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))*
(1 - (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)) -
(2*(-1 + n)*x*(3^(2/3)*(-1 + n^2)*x^2 -
3^(1/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*
(27*(1 + n) + (-1 + n)*x^2)]))^(2/3))^2*
((Sqrt[3]*(-1 + n)*(1 + n)^3*x^2)/Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)] + 2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)])))/
(27*(1 + n)^2*((-1 + n)*x^2*(9*(1 + n)^2 +
Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(5/3)) -
((-1 + n)*x*(-(3^(2/3)*(-1 + n^2)*x^2) +
3^(1/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))*((Sqrt[3]*(-1 + n)*(1 + n)^3*x^2)/
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)] +
2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)])))/(9*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(4/3)) +
((-1 + n)*x^2*(-(3^(2/3)*(-1 + n^2)*x^2) +
3^(1/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))*((Sqrt[3]*(-1 + n)*(1 + n)^3*x^2)/
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)] +
2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)])))/(9*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(4/3)) + (-2*3^(2/3)*(-1 + n^2)*x +
(2*((Sqrt[3]*(-1 + n^2)^2*x^3)/Sqrt[27 + 54*n - x^2 +
n^2*(27 + x^2)] + 2*(-1 + n^2)*x*(9 + 9*n +
Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)])))/
(3^(2/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*
Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)))/
(3*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)) -
(x*(-2*3^(2/3)*(-1 + n^2)*x +
(2*((Sqrt[3]*(-1 + n^2)^2*x^3)/Sqrt[27 + 54*n - x^2 +
n^2*(27 + x^2)] + 2*(-1 + n^2)*x*(9 + 9*n +
Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)])))/
(3^(2/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*
(27*(1 + n) + (-1 + n)*x^2)]))^(1/3))))/
(3*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)) +
(2*(-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))*(-2*3^(2/3)*(-1 + n^2)*x +
(2*((Sqrt[3]*(-1 + n^2)^2*x^3)/Sqrt[27 + 54*n - x^2 +
n^2*(27 + x^2)] + 2*(-1 + n^2)*x*(9 + 9*n +
Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)])))/
(3^(2/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*
(27*(1 + n) + (-1 + n)*x^2)]))^(1/3))))/
(9*(1 + n)^2*((-1 + n)*x^2*(9*(1 + n)^2 +
Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(2/3)) +
n*(-1 + (-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))/(3*(1 + n)*
((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))^(1/3)))*
(1 - ((-1 + n)*x*(-(3^(2/3)*(-1 + n^2)*x^2) +
3^(1/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*
(27*(1 + n) + (-1 + n)*x^2)]))^(2/3))*
((Sqrt[3]*(-1 + n)*(1 + n)^3*x^2)/Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)] + 2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)])))/
(9*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(4/3)) +
(-2*3^(2/3)*(-1 + n^2)*x + (2*((Sqrt[3]*(-1 + n^2)^2*x^3)/
Sqrt[27 + 54*n - x^2 + n^2*(27 + x^2)] + 2*(-1 + n^2)*x*
(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)])))/
(3^(2/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*
(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)))/
(3*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(1/3))) +
((-1 + n)*n*x*(x + (-(3^(2/3)*(-1 + n^2)*x^2) +
3^(1/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*
(27*(1 + n) + (-1 + n)*x^2)]))^(2/3))/
(3*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*
(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)))*
(-((-(3^(2/3)*(-1 + n^2)*x^2) + 3^(1/3)*((-1 + n^2)*x^2*
(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) + (-1 + n)*x^2)]))^
(2/3))*((Sqrt[3]*(-1 + n)*(1 + n)^3*x^2)/
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)] +
2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
(-1 + n)*x^2)]))) + 3*x*(9*(1 + n)^2 +
Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)])*
(-2*3^(2/3)*(-1 + n^2)*x + (2*(-1 + n^2)*x*((Sqrt[3]*(-1 + n^2)*x^2)/
Sqrt[27 + 54*n - x^2 + n^2*(27 + x^2)] +
2*(9 + 9*n + Sqrt[3]*Sqrt[27 + 54*n - x^2 + n^2*(27 + x^2)])))/
(3^(2/3)*((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*
(27*(1 + n) + (-1 + n)*x^2)]))^(1/3)))))/
(9*(1 + n)*((-1 + n)*x^2*(9*(1 + n)^2 + Sqrt[3]*
Sqrt[(1 + n)^3*(27*(1 + n) + (-1 + n)*x^2)]))^(4/3))))/
(2*(3^(2/3)*(-1 + n^2)*x^2 - 3^(1/3)*
((-1 + n^2)*x^2*(9 + 9*n + Sqrt[3]*Sqrt[(1 + n)*(27*(1 + n) +
(-1 + n)*x^2)]))^(2/3))^2)

--- \/   FROM Andrzej Kozlowski AT 05.05.24 10:15 (Today)   \/ ---

> GroebnerBasis[FullSimplify[D[f[x, n], x]], x]
>   also works for n other than 2 and confirms my conjecture.  For
> example;
>
>
> GroebnerBasis[FullSimplify[D[f[x, 3], x]], x]
>
>
> {-9*x^2 - 18*x + 16, 9*x + 18 - 16/x,
>    -18*x - 9*(x^2 + 54) + 502,
>    27*6^(1/3)*Sqrt[x^2 + 54]*x + 35*6^(5/6)*x -
>     68*6^(1/3)*Sqrt[x^2 + 54] + 350*6^(1/6)*
>      (x^2*(Sqrt[6]*Sqrt[x^2 + 54] + 18))^(1/3) -
>     140*6^(5/6)}
>
>
> Solve[-9*x^2 - 18*x + 16 == 0, x]
>
>
> {{x -> -(8/3)}, {x -> 2/3}}
>
> I suspect that this is actually what FindInstance uses (?)
>
>
> So here is a new formulation of the conjecture: the roots of the
> equation D[f[x,n],x]==0 are precisely the roots of the quadratic
> equation
>
>
> x^2 + (n*x)/(n - 1) + (2*x)/(n - 1) + x/((n - 1)*n) +
>    x/n - x - n/(n - 1) - 1/(n - 1) + 1/((n - 1)*n) +
>    1/((n - 1)*n^2) ==0
>
>
> Andrzej Kozlowski
>
>
>
> On 24 May 2005, at 08:11, Daniel Reeves wrote:
>
> > *This message was transferred with a trial version of CommuniGate
> > (tm) Pro*
> > Ooh ooh!  I just got FindInstance to prove it for any particular n.
> > (Tried
> > it for all n up to 75 and it takes about 20 seconds each on my
> > machine and
> > doesn't slow down as n increases, though every once in a while
> > there's an
> > n that takes 2 minutes).
> >
> > n = 7;
> > FindInstance[FullSimplify[D[f[x,n],x]] == 0 && 0<x<(n-1)/n, {x},
> > Reals]
> >   --> returns {} and so QED.
> >
> > Prize for the general case bigger than ever...
> >
> >
> >
> >> Unfortunately I can only manage the most trivial case, n=2, which I
> >> do not think deserves any prize but it seems to show that if anyone
> >> does get the prize it will be well deserved.
> >>
> >> We evaluate your definitions and then set
> >>
> >> n = 2;
> >>
> >> Even in this simplest of cases trying to use Solve directly
> >>
> >>
> >> Solve[D[f[x,2],x]==0,x]
> >>
> >> \$Aborted
> >>
> >> takes for ever, so we resort to Groebner basis (actually what I just
> >> called "a hack" in another discussion ;-))
> >>
> >> We first evaluate
> >>
> >> z = FullSimplify[D[f[x, 2], x] /. x^2 + 81 -> u^2, u > 0];
> >>
> >> And then use GroebnerBasis:
> >>
> >>
> >> GroebnerBasis[{x^2 + 81 - u^2, z}, {x}, {u}]
> >>
> >> {4*x^2 + 16*x - 9}
> >>
> >> This we can actually solve (even without Mathematica):
> >>
> >>
> >> r=Solve[4*x^2 + 16*x - 9 == 0, x]
> >>
> >> {{x -> -(9/2)}, {x -> 1/2}}
> >>
> >> we can check that the derivative of f[x,2] does indeed vanish at
> >> these points:
> >>
> >>
> >> D[f[x,2],x]/.r//FullSimplify
> >>
> >> {0,0}
> >>
> >> Assuming that Groebner basis is correctly implemented this also shows
> >> that f'[x,2] has no roots in the interval (0,1/2) so we are done.
> >>
> >>
> >> Unfortunately the same method with n=3 already seems to take for
> >> ever. For small n>2 it may be possible perhaps to combine in some way
> >> numerical methods with Groebner basis to show similarly that D[f
> >> [x,n],x] has no roots in the interval (0,(n-1)/n) but obviously this
> >> won't work for a general n. At the moment the general problem looks
> >> pretty intractable to me, so much that I am almost willing to offer
> >> an additional prize to whoever manages to solve it (but I won't make
> >> any offers before seeing other people's attempts ;-))
> >>
> >> Andrzej Kozlowski
> >>
> >
> > --
> > http://ai.eecs.umich.edu/people/dreeves  - -  google://"Daniel Reeves"
> >
> > "The good Christian should beware of mathematicians and all those who
> > make empty prophecies.  The danger already exists that mathematicians
> > have made a covenant with the devil to darken the spirit and confine
> > man in the bonds of Hell."
> >                 -- St. Augustine

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