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

On 23 May 2005, at 15:20, Daniel Reeves 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)
> -- 
>  - -  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.

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



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:



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

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