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[[File:Mplwp Steiners problem.svg|right|300px]]
'''Steiner's problem''', asked and answered by {{harvtxt|Steiner|1850}}, is the problem of finding the [[Maxima and minima|maximum]] of the [[Function (mathematics)|function]]

: <math>f(x)=x^{1/x}.\,</math><ref>{{cite web
| url = http://mathworld.wolfram.com/SteinersProblem.html
| title = Steiner's Problem
| author = Eric W. Weisstein
| publisher = MathWorld
|accessdate=December 8, 2010 }}</ref>

It is named after [[Biography:Jakob Steiner|Jakob Steiner]].

The maximum is at <math>x = e</math>, where ''e'' denotes the [[E (mathematical constant)|base of the natural logarithm]]. One can determine that by solving the equivalent problem of maximizing 

: <math>g(x) = \ln f(x) = \frac{\ln x}{x}.</math>

Applying the first derivative test, the [[Derivative|derivative]] of <math>g</math> is

: <math>g'(x) = \frac{1-\ln x}{x^2},</math>

so <math>g'(x)</math> is positive for <math>0<x<e</math> and negative for <math>x>e</math>, which implies that <math>g(x)</math> – and therefore <math>f(x)</math> – is increasing for <math>0<x<e</math> and decreasing for <math>x>e.</math> Thus, <math>x=e</math> is the unique [[Maxima and minima#Definition|global maximum]] of <math>f(x).</math>

==References==
<references />
*{{citation|title=Über das größte Product der Theile oder Summanden jeder Zahl
|first=J. |last=Steiner|journal=Crelle|volume=40|year=1850|pages=208|url=https://www.digizeitschriften.de/download/PPN243919689_0040/PPN243919689_0040___log28.pdf}}

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