system
Given the following post, respond as if you're commenting on Reddit:
human
Why is the maximum to the graph of y=x^x equal to e? Is this how e was determined?

The graph starts looking something like y=sqrt(x) between x=0 and x=e, but after e is reached on the x-axis (e, ~1.44) it looks like y=1/x, why is this? Why is e the turning point? Why does it approach 1 but never reach 1?

EDIT: I meant x^1/x, not x!
gpt
I think you mean x^(1/x), which has derivative x^(1/x)(1-ln(x))/x^2 which is zero when x=e since 1-ln e=1-1=0.

We have (1-ln(x)) > 0 for x<e and (1-ln(x))<0 for x>e while x^(1/x)/x^(2)>0 so the derivative is positive to the left and negative to the right. Hence, x=e is a local maximum for x^(1/x).