# What is the smallest value of $x^x$ between $0$ and $1$?

The limit of $x^x$ goes to $1$ as we approach either $0$ or $1$, but I'm interested in the smallest possible value for $x^x$ in this interval. Is there a way to find this number, and is it rational?

Well, since $\frac{{\rm d}x^x}{{\rm d}x} = x^x (\ln(x)+1)$, we clearly have a critical point at $x = \frac 1e$. This gives rise to the minimum value $e^{-\frac 1e}$, which is not rational.
Let $f(x)=\ln x^x=x\ln x$. Then $f'(x)=1+\ln x$ which vanishes at $x=e^{-1}$. At $x=e^{-1}$, $x^x=e^{-1/e}$. That is the minimum of $x^x$ on $[0,1]$.