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.