$x^y+y^x>1$ for all $(x, y)\in \mathbb{R_+^2}$

Question

Prove that $x^y+y^x>1$ for all $(x, y)\in \mathbb{R_+^2}$.

Answer 1

use This $(1+x)^a<1+ax,0<a<1,x>0$

then this problem we only prove $0<x<1,0<y<1 $

$$x^y=\dfrac{1}{(\dfrac{1}{x})^y}=\dfrac{1}{(1+\dfrac{1-x}{x})^y}>\dfrac{1}{1+\dfrac{(1-x)y}{x}}=\dfrac{x}{x+y-xy}>\dfrac{x}{x+y}$$

and $$y^x>\dfrac{y}{x+y}$$