Exponential inequality(Proof)

Question

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).

Answer 1

Case I: $x$ and $y$ both greater than 1
$x^y, y^x > 1$
Hence, proved

Case II: x is 1 (without loss of generality)
$x^y$ = 1
$1 + y^x > 1$
Hence, proved

Case III: $x$ and $y$ both less than 1.
I can explain why this could be true informally. For both x and y less than 1, the power is actually a root, and a root of a number less than 1 will increase its value. Still, can't show that it will be at least one.

Answer 2

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!