(a&sup4+b&sup4)(a²+b²)与(a³+b³)²

(a^4+b^4)(a^2+b^2)-(a^3+b^3)^2 =(a^6+b^6+a^4b^2+a^2b^4)-(a^6+b^6+2a^3b^3) =a^4b^2+a^2b^4-2a^3b^2 =a^2b^2(a^2+b^2-2ab) =a^2b^2(a-b)^2 ∵a^2>=0,b^2>=0,(a-b)^2>0 ∴(a^4+b^4)(a^2+b^2)-(a^3+b^3)^2=a^2b^2(a-b)^2>=0 ∴(a^4+b^4)(a^2+b^2)>=(a^3+b^3)^2(取等a=0或b=0)