Editing E (mathematical constant) (section)

=== Optimal planning problems ===
The maximum value of <math>\sqrt[x]{x}</math> occurs at <math>x = e</math>.  Equivalently, for any value of the base {{math|''b'' > 1}}, it is the case that the maximum value of <math>x^{-1}\log_b x</math> occurs at <math>x = e</math> ([[Steiner's calculus problem|Steiner's problem]], discussed [[#Exponential-like functions|below]]).

This is useful in the problem of a stick of length {{mvar|L}} that is broken into {{mvar|n}} equal parts.  The value of {{mvar|n}} that maximizes the product of the lengths is then either<ref name="Finch-2003-p14">{{cite book|title=Mathematical constants|url=https://archive.org/details/mathematicalcons0000finc|url-access=registration|author=Steven Finch|year=2003|publisher=Cambridge University Press|page=[https://archive.org/details/mathematicalcons0000finc/page/14 14]|isbn=978-0-521-81805-6}}</ref>
:<math>n = \left\lfloor \frac{L}{e} \right\rfloor</math> or <math>\left\lceil \frac{L}{e} \right\rceil.</math>

The quantity <math>x^{-1}\log_b x</math> is also a measure of [[Shannon information|information]] gleaned from an event occurring with probability <math>1/x</math> (approximately <math>36.8\%</math> when <math>x=e</math>), so that essentially the same optimal division appears in optimal planning problems like the [[secretary problem]].