TITLE: Proof of Inequality using AM-GM
QUESTION [7 upvotes]: I just started doing AM-GM inequalities for the first time about two hours ago. In those two hours, I have completed exactly two problems. I am stuck on this third one! Here is the problem:
If $a, b, c \gt 0$ prove that $$ a^3 +b^3 +c^3 \ge a^2b +b^2c+c^2a.$$
I am going crazy over this! A hint or proof would be much appreciated. Also any general advice for proving AM-GM inequalities would bring me happiness to my heart. Thank you!

REPLY [3 votes]: $$\color{blue}{a^2(a-b)+b^2(b-c)+c^2(c-a)=(a^2-c^2)(a-b)+(b^2-c^2)(b-c)}$$
$$\color{red}{a\geq b \geq c \implies (a^2-c^2)(a-b)+(b^2-c^2)(b-c) \geq0}$$