(2/3)比较(a^4)+(b^4)与(a^3)b+a(b^3)的大小为什么是当a不等于b时,则有(a^4)+(b^4)>(a^3)b+a(b^

来源:学生作业帮助网 编辑:作业帮 时间:2024/11/06 08:09:03
(2/3)比较(a^4)+(b^4)与(a^3)b+a(b^3)的大小为什么是当a不等于b时,则有(a^4)+(b^4)>(a^3)b+a(b^
xJ@F]dv@ER jAzZSF6Io}'IL[If9i6Mfm o9bfShm/J\~YrBrnd2. m9|r9?^Ehy(tXc \ХE" 4 h7He:OgNQ8NJfj^jΟ26e$)`/x(6k*& EXH К=YS@[Ng6uJ7SnVsɞQ|

(2/3)比较(a^4)+(b^4)与(a^3)b+a(b^3)的大小为什么是当a不等于b时,则有(a^4)+(b^4)>(a^3)b+a(b^
(2/3)比较(a^4)+(b^4)与(a^3)b+a(b^3)的大小为什么是当a不等于b时,则有(a^4)+(b^4)>(a^3)b+a(b^

(2/3)比较(a^4)+(b^4)与(a^3)b+a(b^3)的大小为什么是当a不等于b时,则有(a^4)+(b^4)>(a^3)b+a(b^
作差:[a^4+b^4]-[a³b+ab³]=[a^4-a³b]+[b^4-ab³]=a³(a-b)+b³(b-a)=(a-b)(a³-b³)
=(a-b)²(a²+ab+b²)=(a-b)²{[a+(b/2)]²+(3/4)b²}
由于(a-b)²≥0,[a+(b/2)]²+(3/4)b²>0,成立

(a^4)+(b^4)-(a^3)b+a(b^3)
=(a^3)(a-b)-(b^3)(a-b)
=(a-b)[(a^3)-(b^3)]
=A
若a>b,a-b>0;(a^3)-(b^3)>0,A>0,(a^4)+(b^4)>(a^3)b+a(b^3);
若a0,(a^4)+(b^4)>(a^3)b+a(b^3);