<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] The inequality is equivalent to ( by expanding RHS) 
$$ x^4 + y^4 +z^4 \ge x^2yz + y^2xz + z^2xy$$
By AM-GM
$$ x^4+x^4 +y^4 +z^4 \ge 4(x^8y^4z^4)^\frac{1}{4} = 4x^2yz$$
$$ y^4+y^4 +x^4 +z^4 \ge 4(y^8x^4z^4)^\frac{1}{4} = 4y^2xz$$
$$ z^4+z^4 +x^4 +y^4 \ge 4(z^8x^4y^4)^\frac{1}{4} = 4z^2xy$$
Adding them up we get
$$4(x^4 +y^4 +z^4) \ge 4(x^2yz+y^2xz + z^2xy)$$
$$ x^4 +y^4 +z^4 \ge x^2yz+y^2xz + z^2xy $$
</s>