system
Given the following post, respond as if you're commenting on Reddit:
human
Why is a turning point on the plot of 1^1 to 0^0 ?

So we take the fallowing problem :

1^1 = 1

0.9^0.9 = 0.909

0.8^0.8 = 0.836

0.7^0.7 = 0.779

0.6^0.6 = 0.736

0.5^0.5 = 0.707

0.4^0.4 = 0.693

0.3^0.3 = 0.696

0.2^0.2 = 0.724

0.1^0.1 = 0.793

0.0^0.0 ~ 1

So we see that somewhere between 0.5^0.5 and 0.3^0.3 it will be the lowest value. Seems that the turning point is 1/e ( 0.36787944117^0.36787944117 = 0.69220062755 ). Why there ? Is there a special thing about this number ? Does this ”problem” have a name ?

Sorry if this a noob question, but it's the first time I see this pattern.
gpt
If you know calculus, you can compute it easy. We want to find the minimum of x^(x). If we take natural log to get f(x)=xlog(x), then the minimum of xlog(x) will be at the same point as x^(x), so we can work with f(x) instead. Compute the derivative to be f'(x)=log(x)+1. We then know that if f'(x)=0, then we have an extrema. Solving log(x)+1=0 will give us x=1/e. This can easily be checked to be a minimum point using the second derivative f''(x)=1/x, for which we have f''(1/e)>0. Therefore (1/e)^(1/e) is the minimum of x^(x).