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Re: how to show for what values the function is increasing/decreasing

  • To: mathgroup at smc.vnet.net
  • Subject: [mg104102] Re: how to show for what values the function is increasing/decreasing
  • From: pfalloon <pfalloon at gmail.com>
  • Date: Mon, 19 Oct 2009 07:12:26 -0400 (EDT)
  • References: <hbemj1$gfj$1@smc.vnet.net>

On Oct 18, 8:19 pm, JEZUS <barefoot1... at gmail.com> wrote:
> how to show
>
> that for what values of m, the function
>
> f(x) = m * log(x) / 2^m + (1-x^m) / (1+x)^m
>
> is increasing/decreasing. That for what values of m, df/dx > 0 for all
> x>=1, ....
>
> here, x >= 1
>
> it looks like the (i am not sure):
>
> df/dx > 0 for 0 < m <=3
>
> df/dx < 0 for m < 0
>
> df/dx < 0 for m > 0

Here is a fairly simple-minded approach to the problem using
Mathematica:

    (* clear any previous definitions *)
    Clear[f,m,x]

    (* define function of m and x *)
    f[m_,x_] = m * Log[x] / 2^m + (1-x^m) / (1+x)^m

   (* visualize as a function of x for different *)
   Manipulate[Plot[f[m,x], {x,1,100}, PlotRange->All, AxesOrigin->
{1,0}], {m,0.001,10,Appearance->"Labeled"}]

At this point, playing around with slider control for m shows that for
m less than approximately 3, the curve increases monotonically for
x>1, while for larger m it is no longer monotonic and starts off as a
decreasing function.

This suggests that we look at the behaviour of the function and its
derivatives near x == 1:

    (* evaluate function and derivatives at x=1 *)
    Table[FullSimplify[Derivative[0,n][f][m,1]], {n,0,3}] // TableForm

    0
    0
    0
    -2^(-2-m) (-3+m) m^2

This shows that the first three derivatives vanish at x==1. So the
sign of the third derivative at x==1 will tell us whether f[m,x] is
going to increase or decrease for x>1:

    (* find zeros of third derivative *)
   Reduce[Derivative[0,3][f][m,1] == 0, m]

    m==0 || m==3

So the cutoffs are 0 and 3 as you noted.

Hope this is useful.

Cheers,
Peter.


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