<s>[INST] Proving the exponential inequality: $x^y+y^x\gt1$ How can the following inequality be proven?
$$x^y+y^x\gt1$$
for $(x\gt0,y\lt1)$
 [/INST] Use this Bernoulli inequality
$$(1+x)^a\le 1+ax,0<a<1,x>-1$$
It is clear we only prove $0<x,y<1$
then 
$$x^y=\dfrac{1}{\left(\dfrac{1}{x}\right)^y}=\dfrac{1}{\left(1+\dfrac{1-x}{x}\right)^y}>\dfrac{1}{1+\dfrac{1-x}{x}\cdot y}=\dfrac{x}{x+y-xy}>\dfrac{x}{x+y}$$
Similarly 
$$y^x>\dfrac{y}{x+y}$$
so
$$x^y+y^x>\dfrac{x}{x+y}+\dfrac{y}{x+y}=1$$
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