Let a,b,c be real numbers. Is a,b,c≥0 the necessary and sufficient condition to show that a³+b³+c³≥3abc? If not, find the condition that is both sufficient and necessary.

Well, it is certainly not necessary.
Consider (a,b,c) = (-1, 1, 1).
a³+b³+c³ > 3abc, even though it is not the case that a,b,c≥0

Posted by Steve Herman on 2008-02-27 11:34:08