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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

duckegg
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    That minimum is at $x=1/e$. – Gerry Myerson Jun 30 '22 at 11:09
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    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 $x<1/e$. – wasn't me Jun 30 '22 at 13:46
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    I've deleted some comments. Please keep interactions civil and avoid long exchanges that are not relevant to the post at hand. – Pedro Jul 02 '22 at 21:14

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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.

duckegg
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