<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] AM–GM is invoked in two steps as follows:
$\begin{array}{rcl}x^4+y^4+z^4 &=& \dfrac{x^4+y^4}2+\dfrac{y^4+z^4}2+\dfrac{z^4+x^4}2 \\\\ &\ge& x^2y^2+y^2z^2+z^2x^2 \\\\ &=& \dfrac{x^2y^2+y^2z^2}2+\dfrac{y^2z^2+z^2x^2}2+\dfrac{z^2x^2+x^2y^2}2 \\\\  &\ge& xy^2z+yz^2x+zx^2y \\\\ &=& xyz(x+y+z)\end{array}$
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