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I've noticed that x^x is a minimum for x = e^-1

I put it as a high school problem because I presume it's one of those simple differential proofs/identities, but I can't really see how to get to e^-1. Too long since I did any calculus. Can someone please show me how to arrive at that?

How about (x^x)^x is a minimum for x = e^-(1/2)?

and ((x^x)^x)^x is a minimum for x = e^-(1/3)

I presume the pattern goes on.

I put it as a high school problem because I presume it's one of those simple differential proofs/identities, but I can't really see how to get to e^-1. Too long since I did any calculus. Can someone please show me how to arrive at that?

How about (x^x)^x is a minimum for x = e^-(1/2)?

and ((x^x)^x)^x is a minimum for x = e^-(1/3)

I presume the pattern goes on.

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