Question

Prove that, for all x, y, z >_0, x2 + xy2 + xyz2 >_4xyz − 4.

Answer 1

If x \geq y > 0, then x^ {2} \geq yx \geq y^ {2}; so x^ {2} < y^ {2} implies x<y. For a why you did not do it right, check out the first comment below. If x ≥ y > 0, then x2 ≥ yx ≥ y2; so x2 < y2 implies x < y. For a why you did not do it right, check out the first comment below.