(a²+b²+2)/2-(a²-2b²+1)/3
=[3(a²+b²+2)-2(a²-2b²+1)]/6
=(3a²+3b²+6-2a²+4b²-2)/6
=(a²+7b²+4)/6>0 ∵a²+7b²+4>0
∴(a²+b²+2)/2>(a²-2b²+1)/3
(a²+b²+2)/2-(a²-2b²+1)/3
=[3(a²+b²+2)-2(a²-2b²+1)]/6
=(3a²+3b²+6-2a²+4b²-2)/6
=(a²+7b²+4)/6>0 ∵a²+7b²+4>0
∴(a²+b²+2)/2>(a²-2b²+1)/3