Steiner's Problem

Author

Eric W. Weisstein
February 7, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/SteinersProblem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SteinersProblem.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Maximum

<<Utilities`Plot`
<<Utilities`Typesetting`

PlotDiscontinuous[x^(1/x),{x,0,15},PlotStyle->Red,
HorizontalAsymptotes->{1},AxesLabel->TraditionalForm/@{x,x^InlineFraction[1,x]}]

Simplify[D[x^(1/x),x]]

Solve[x^(-2+1/x)(1-Log[x])==0,x]

Maximize[{x^(1/x),x>0},x]

Limit

Limit[(x^(1/x)),x->∞]

Inflection Point

FullSimplify[D[x^(1/x),{x,2}]]

f[x_?NumericQ]:=(x^(-4+1/x)) (1-3 x+Log[x] (-2+2 x+Log[x]))

Plot[f[x],{x,0,5},PlotRange->{-.05,.05},PlotStyle->Red]

Reduce[(x^(-4+(1/x))) (1-3 x+Log[x] (-2+2 x+Log[x]))==0,x,Reals]

infl1=x/.FindRoot[f[x]==0,{x,.6},WorkingPrecision->110]

infl2=x/.FindRoot[f[x]==0,{x,4.5},WorkingPrecision->110]