(a) A function $f$ is defined by $f(x)=x^x$ for all $x>0$. Find the minimum value of $f$. (b) If $x$ and $y$ are two positive real numbers, show that $x^y+y^x>1$.
Source: French MO 1996 P4
Tags: Inequality, optimization, inequalities
(a) A function $f$ is defined by $f(x)=x^x$ for all $x>0$. Find the minimum value of $f$. (b) If $x$ and $y$ are two positive real numbers, show that $x^y+y^x>1$.