TITLE: derivatives of $x^x$ positive on a halfline
QUESTION [6 upvotes]: I'm being curious how it is:
Is there a number $a$ such that all derivatives of $x^x$ are positive on $(a,\infty)$ ?

REPLY [3 votes]: This question was proposed by Henson, Reznick and Rubel in The American Mathematical Monthly, Vol. 91, No. 1 (Jan., 1984), page 60. The answer was given by Chernoff (and independently by others). In short, if such $a$ exists, then the assumptions and the Taylor expansion theorem give that the Taylor series for $x^x$ converges and has an infinite radius of convergence. This implies that $x^x$ is an entire function, which is a contradiction.
Another reference is Biler, Witkowski "Problems in Mathematical Analysis" (Chapman & Hall/CRC Pure and Applied Mathematics), see problem 4.215.