I am struggling with learning the AM-GM Inequality that is considered vital to know for math olympiads, contests, etc. I just don't really know when to use it, when it is necessary to use, the purpose of it, etc...
I know,
$$\frac{x_1 + x_2 + \dots +x_n}{ n} \ge (x_1x_2\cdots x_n)^{1/n}$$
I've watched a few videos on using the inequality for optimization problems and all, but still can't grasp it completely enough. I feel like either my understanding is still relatively poor to try a few problems, or that I'm missing a few prerequisites (a basic understanding of number theory, or perhaps other inequality concepts?)
I've watched this video, which kind of helped a bit. If someone could suggest some learning resources, or books regarding the AM-GM Inequality, that would be seriously wonderful.
Thanks so much.
EDIT: Having trouble with this part of Cauchy-Schwarz Master Class book.
It is written that $$xy \le \frac{x^2 }{2} + \frac{y^2 }{ 2}$$ which is correct. The author proceeds to say that this is a bit of a trivial statement, and that in order to extract more information from it, we replace x and y with their square roots. He then gets $4\sqrt{xy} \le 2x + 2y$.
I don't understand how he got that. The closest I got to it so far was,
$$xy \le (x^2/2) + (y^2/2)$$
$$2\sqrt{xy} \le (\sqrt{x}-\sqrt{y})^2 + 2xy$$
getting,
$$2\sqrt{xy} \le x + y$$
while he gets,
$$4\sqrt{xy} \le 2x + 2y$$
I understand that his is just twice bigger on both sides, but what steps did he take to get to that? I don't understand what he did in general. Would be awesome if someone could guide me through this.
Thanks!