已知a,b,c>0,求证a³+b³+c³≧1/3﹙a²+b²+c²﹚﹙a+b+c﹚
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已知a,b,c>0,求证a³+b³+c³≧1/3﹙a²+b²+c²﹚﹙a+b+c﹚
已知a,b,c>0,求证a³+b³+c³≧1/3﹙a²+b²+c²﹚﹙a+b+c﹚
已知a,b,c>0,求证a³+b³+c³≧1/3﹙a²+b²+c²﹚﹙a+b+c﹚
3(a^3+b^3+c^3)-(a^2+b^2+c^2)(a+b+c)
=3(a^3+b^3+c^3)-(a^3+b^3+c^3+a^2b+a^2c+ab^2+b^2c+ac^2+bc^2)
=2a^3+2b^3+2c^3-a^2b-a^2c-ab^2-b^2c-ac^2-bc^2
=(a^3+b^3-a^2b-ab^2)+(b^3+c^3-bc^2-b^2c)+(c^3+a^3-ca^2-ac^2)
=(a+b)(a^2-2ab+b^2)+(b+c)(b^2+c^2-2bc)+(a+c)(a^2+c^2-2ac)
=(a+b)(a-b)^2+(b+c)(b-c)^2+(a+c)(a-c)^2≥0
∴a^3+b^3+c^3≥1/3(a^2+b^2+c^2)(a+b+c)