Multi-compartment model: Difference between revisions
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[[File:Steiner problem.svg|right|300px]] | |||
'''Steiner's problem''' 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 = 12/08/2010 | |||
}}</ref> | |||
It is named after [[Jakob Steiner]]. | |||
The maximum is at <math>x=e</math>, where ''e'' denotes the [[e (mathematical constant)|base of natural logarithms]]. One can determine that by solving the equivalent problem of maximizing | |||
: <math>g(x)=\ln f(x) = \frac{\ln x}{x}.</math> | |||
The [[derivative]] of <math>g</math> can be calculated to be | |||
: <math>g'(x)= \frac{1-\ln x}{x^2}.</math> | |||
It follows that <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>) increases for <math>0<x<e</math> and decreases for <math>x>e.</math> Thus, <math>x=e</math> is the unique global maximum of <math>f(x).</math> | |||
==References== | |||
<references /> | |||
{{DEFAULTSORT:Steiner's Problem}} | |||
[[Category:Functions and mappings]] | |||
[[Category:Mathematical optimization]] | |||
Revision as of 15:34, 6 December 2013
Steiner's problem is the problem of finding the maximum of the function
It is named after Jakob Steiner.
The maximum is at , where e denotes the base of natural logarithms. One can determine that by solving the equivalent problem of maximizing
The derivative of can be calculated to be
It follows that is positive for and negative for , which implies that (and therefore ) increases for and decreases for Thus, is the unique global maximum of
