Question

设a＞0,b＞0,试比较√（a^3+b^3）与√（a^2+b^2）的大小

Answer 1

设x=(√（a^3+b^3）)^6-()√（a^2+b^2）^6 =(a^3+b^3)^2-(a^2+b^2)^3 =a^6+2a^3b^3+b^6-(a^6+3a^4b^2+3a^2b^4+b^6) =2a^3b^3-3(a^4b^2+a^2b^4) 因为a^2+b^2>=2ab 所以a^4b^2+a^2b^4>=2a^3b^3 a＞0,b＞0 3(a^4b^2+a^2b^4)>=3*2a^3b^3>=2a^3b^3 x