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Re: how to show for what values the function is


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



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