TITLE: Exponential inequality(Proof)
QUESTION [8 upvotes]: Given $\forall x,y>0$
Prove that $x^y+y^x\ge1$  
I have tried weighted inequalities and Jensen's but unfortunately ended up no where.
Please help me. (I know this is a basic inequality).

REPLY [2 votes]: For $x\geq1$ or $y\geq1$ the inequality is obviously true.
But for $\{x,y\}\subset(0,1)$ easy to show that $x^y\geq\frac{x}{x+y}$ 
and we are done!