Re: how to show for what values the function is

*To*: mathgroup at smc.vnet.net*Subject*: [mg104099] Re: how to show for what values the function is*From*: Bob Hanlon <hanlonr at cox.net>*Date*: Mon, 19 Oct 2009 07:11:51 -0400 (EDT)*Reply-to*: hanlonr at cox.net

f[x_, m_] = m*Log[x]/2^m + (1 - x^m)/(1 + x)^m; dfdx[x_, m_] = FullSimplify[D[f[x, m], x]]; Plot3D[f[x, m], {x, 1, 20}, {m, -2, 5}, PlotRange -> {-1/2, 1}, ClippingStyle -> None, ColorFunction -> Function[{x, m, z}, If[Re[dfdx[x, m]] > 0, Lighter[Blue, .6], Red]], ColorFunctionScaling -> False, Mesh -> 10, PlotPoints -> 75] Plot3D[{dfdx[x, m], 0}, {x, 1, 20}, {m, -2, 5}, PlotRange -> {-1/16, 1/16}, ClippingStyle -> None, PlotStyle -> {Automatic, Directive[Red, Opacity[.75]]}] ContourPlot[dfdx[x, m], {x, 1, 20}, {m, -2, 5}] Bob Hanlon ---- JEZUS <barefoot1980 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