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| [[File:Steiner problem.svg|right|300px]]
| | Jayson Berryhill is how I'm known as and my wife doesn't like it at all. The favorite hobby for him and his children is to perform lacross and he would by no means give it up. Mississippi is where his house is. Credit authorising is where my primary earnings comes from.<br><br>Feel free to surf to my site: online psychic ([http://brazil.amor-amore.com/irboothe one-time offer]) |
| '''Steiner's problem''' is the problem of finding the [[maxima and minima|maximum]] of the [[function (mathematics)|function]]
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| : <math>f(x)=x^{1/x}.\,</math><ref>{{cite web
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| | url = http://mathworld.wolfram.com/SteinersProblem.html
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| | title = Steiner's Problem
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| | author = Eric W. Weisstein
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| | publisher = MathWorld
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| | accessdate = 12/08/2010
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| }}</ref>
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| It is named after [[Jakob Steiner]].
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| 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
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| : <math>g(x)=\ln f(x) = \frac{\ln x}{x}.</math>
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| The [[derivative]] of <math>g</math> can be calculated to be
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| : <math>g'(x)= \frac{1-\ln x}{x^2}.</math> | |
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| 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>
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| ==References==
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| <references />
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| {{DEFAULTSORT:Steiner's Problem}}
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| [[Category:Functions and mappings]]
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| [[Category:Mathematical optimization]]
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Latest revision as of 13:46, 5 May 2014
Jayson Berryhill is how I'm known as and my wife doesn't like it at all. The favorite hobby for him and his children is to perform lacross and he would by no means give it up. Mississippi is where his house is. Credit authorising is where my primary earnings comes from.
Feel free to surf to my site: online psychic (one-time offer)