Does there exist an application that makes use of the degenerate case of $x^x$ in the unit range?

For most of the function $x^x$ the y value's progression as $x$ increases or decreases makes sense. But within the unit range (specifically at the extrema 0.3679...) something non-intuitive occurs. I find this inherently interesting but I've been challenged to provide an application for the phenomenon.

Are there any?

degenerate case of x^x

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    $\begingroup$ That minimum is at $x=1/e$. $\endgroup$ Commented Jun 30, 2022 at 11:09
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    $\begingroup$ Why would that be non-intuitive? $\lim_{x\to0+}x^x=1^1=1$, and $0.25^{0.25}=0.5^{0.5}$, so I'd expect a minimum of the function $f(x)=x^x$ between those values. The derivative is $(1+\ln x)\,x^x$, so it's clearly decreasing for $x<1/e$. $\endgroup$
    – wasn't me
    Commented Jun 30, 2022 at 13:46
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    $\begingroup$ I've deleted some comments. Please keep interactions civil and avoid long exchanges that are not relevant to the post at hand. $\endgroup$
    – Pedro
    Commented Jul 2, 2022 at 21:14

1 Answer 1


Following Gerrys hint, it seems as if my question relates to this post What Are The Uses of Eulers Number in which real world examples of actual applications are given.

Thanks to Gerry and @TheBluegrassMathmatician for answering this one.


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