设a>0,b>0,c>0,a≠b,b≠c,c≠a,且a,b,c满足a+b>c,求证：a^3+b^3+c^3+3abc>2(a+b)c^2

设a>0,b>0,c>0,a≠b,b≠c,c≠a,且a,b,c满足a+b>c,求证：a^3+b^3+c^3+3abc>2(a+b)c^2
设a>0,b>0,c>0,a≠b,b≠c,c≠a,且a,b,c满足a+b>c,求证：a^3+b^3+c^3+3abc>2(a+b)c^2

设a>0,b>0,c>0,a≠b,b≠c,c≠a,且a,b,c满足a+b>c,求证：a^3+b^3+c^3+3abc>2(a+b)c^2
a^3+b^3+c^3+3abc
=(a+b)(a^2-ab+b^2)+c^3+3abc
>c(a^2-ab+b^2)+c^3+3abc
=c(a^2-ab+b^2+c^2+3ab)
=c(a^2+2ab+b^2+c^2)
=c[(a+b)^2+c^2]
>=c*2√[(a+b)^2*c^2]
=2(a+b)c^2