Why does X to the power of X decrease for decimals until about X = 0.4 and then it starts to increase again? Why is the location of the turning point where it is?

Question

I’ve recently seen a video where someone did an explanation on why 0 to the power of 0 is defined as 1. He went with X to the power of X and made X increasingly smaller decimals. X = 0.9, X = 0.8, and so on. The results for 0.9 to the power of 0.9 and then 0.8 to the power of 0.8 kept getting smaller until about X = 0.4, where it started to increase again, so 0.3 to the power of 0.3 was a bigger number than 0.4 to 0.4. At what number is the exact “turning point” in this kind of series and why is it in such a weird place (as opposed to at 0.5, i.e halfway to 0)?

In: 5

Answer 1

That turning point is at x = 1 / e
This e is Euler’s number, about 2.71828

*Why* is it 1/e? Because e is everywhere.