Why $e^x$ is always greater than $x^e$? I find it very strange that $$ e^x \geq x^e \, \quad \forall x \in \mathbb{R}^+.$$ 
I have scratched my head for a long time, but could not find any logical reason. Can anybody explain what is the reason behind the above inequality? I know this is math community, but I'd appreciate a more intuitive explanation than a technical one. 

The reasons is that the exponential function($something^x$) grows much faster than the power function ($x^{something}$) 
So in general you would expect  that for every $a, b > 1$something like this hold 
$$a^x \ge x^b$$ at least for $x$ big enough 
Now I don't think it's particularly meaningful that for the special case $a = b = e$ the inequality holds for every $x \ge 0$. It can be shown easily with basic calculus in any case, but maybe that won't satisfy your intuition :)
