<s>[INST] Using AM-GM inequality to prove prove that $$x^4 + y^4 + z^4 \geq xyz(x+y+z)$$
This AM-GM inequalities are seriously stumping me. I'd appreciate a full proof and explanation and hints for proving other inequalities like this. Thanks.
 [/INST] Use: $$a^2+b^2+c^2\ge ab+bc+ca$$
Proof:$$2(a^2+b^2+c^2)\ge 2(ab+bc+ca) \Leftrightarrow$$
$$\Leftrightarrow(a-b)^2+(b-c)^2+(c-a)^2\ge0$$
Then 
$$x^4+y^4+z^4\ge x^2y^2+y^2z^2+z^2x^2\ge xy^2z+x^2yz+xyz^2=$$
$$=xyz(x+y+z)$$
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