If x and y are positive real numbers, show that $x^y+y^x≥1.$

Question

If $x$ and $y$ are positive real numbers, show that $x^y + y^x ≥ 1$.

I think it is trivial that the inequality holds when $x ≥ 1$ or $y ≥ 1$. I'm struggling to prove the inequality when $x < 1$ and $y < 1$.

Answer 1

hint: $$x^y=\frac{1}{\left(\frac{1}{x}\right)^y}=\frac{1}{\left(1+\frac{1-x}{x}\right)^y}\geq \frac{1}{1+\left(\frac{1-x}{x}\right)y}=\frac{x}{x+y-xy}>\frac{x}{x+y}$$ and analogously $$y^x>\frac{y}{x+y}$$