Editing E (mathematical constant) (section)

=== Exponential-like functions ===
[[File:Xth root of x.svg|thumb|right|250px|The [[global maximum]] of {{math|{{sqrt|''x''|''x''}}}} {{nowrap|occurs at {{math|''x'' {{=}} ''e''}}.}}]]

[[Steiner's calculus problem|Steiner's problem]] asks to find the [[global maximum]] for the function

<math display="block"> f(x) = x^\frac{1}{x} .</math>

This maximum occurs precisely at {{math|''x'' {{=}} ''e''}}.  (One can check that the derivative of {{math|ln ''f''(''x'')}} is zero only for this value of&nbsp;{{mvar|x}}.)

Similarly, {{math|''x'' {{=}} 1/''e''}} is where the [[global minimum]] occurs for the function

<math display="block"> f(x) = x^x .</math>

The infinite [[tetration]]

:<math> x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} </math> or <math>{^\infty}x</math>

converges if and only if {{math|''x'' ∈ [(1/''e'')<sup>''e''</sup>, ''e''<sup>1/''e''</sup>] ≈ [0.06599, 1.4447] }},<ref>{{Cite OEIS|A073230|Decimal expansion of (1/e)^e}}</ref><ref>{{Cite OEIS|A073229|Decimal expansion of e^(1/e)}}</ref> shown by a theorem of [[Leonhard Euler]].<ref>Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." ''Acta Acad. Scient. Petropol. 2'', 29–51, 1783. Reprinted in Euler, L. ''Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae''. Leipzig, Germany: Teubner, pp. 350–369, 1921. ([http://math.dartmouth.edu/~euler/docs/originals/E532.pdf facsimile])</ref><ref>{{Cite journal |last=Knoebel |first=R. Arthur |date=1981 |title=Exponentials Reiterated |url=https://www.jstor.org/stable/2320546 |journal=The American Mathematical Monthly |volume=88 |issue=4 |pages=235–252 |doi=10.2307/2320546 |jstor=2320546 |issn=0002-9890}}</ref><ref>{{Cite journal |last=Anderson |first=Joel |date=2004 |title=Iterated Exponentials |url=https://www.jstor.org/stable/4145040 |journal=The American Mathematical Monthly |volume=111 |issue=8 |pages=668–679 |doi=10.2307/4145040 |jstor=4145040 |issn=0002-9890}}</ref>