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Steiner's Problem

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SteinersProblem

For what value of x is f(x)=x^(1/x) a maximum? The maximum occurs at x=e, where

f^'(x)=x^(-2+1/x)(1-lnx)=0,
(1)

which is zero at x=e and gives a maximum of

e^(1/e)=1.444667861...
(2)

(OEIS A073229).

The function has inflection points at x=0.581933... (OEIS A093157) and x=4.36777... (OEIS A103476), which are the roots of

f^('')(x)=x^(-4+1/x)[1-3x+(lnx)(2x-2+lnx)]=0.
(3)

See also

e, Fermat's Problem, MRB Constant, Power Tower

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References

Dörrie, H. 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, 1965.Sloane, N. J. A. Sequences A073229, A093157, and A103476 in "The On-Line Encyclopedia of Integer Sequences."Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 35, 1986.

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Steiner's Problem

Cite this as:

Weisstein, Eric W. "Steiner's Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SteinersProblem.html

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