Let $x$ , $y$ be positive real numbers. Prove the inequality $$x^ y + y^x \ge 1$$
This is the solution provided by my textbook:
Where does this first idea (proving that $a^b \ge \frac{a}{a+ b - ab}$) come from? I'm repeatedly frustrated by problems like these because the solution uses some forlorn idea which seemingly appears out of the blue. I have no idea how to prove such results on my own, because I don't know the steps or methodology required to come up with these intermediate ideas from which the rest of the proof is built. My textbook also unfortunately does not shed any light on the methodology or intuition. If someone who has experience with such methods of problem solving could shed some light on this matter, I would greatly appreciate it.
P.S I'm trying to bring myself to a level where I can take part in math competitions, and my imagination and intuition (or lack thereof) are repeatedly holding me back.
