不等式证明题,求详解已知a、b、c都是实数且a+b+c=1求证根号3a+1加上根号3b+1加上根号3c+1小于等于3倍根号2

Question

不等式证明题,求详解
已知a、b、c都是实数且a+b+c=1求证根号3a+1加上根号3b+1加上根号3c+1小于等于3倍根号2

Answer 1

证：由公式a平方+b平方大于等于2ab得：
(3a+1)+(3b+1)大于等于2根号下（3a+1）*(3b+1)
依此类推：(3a+1)+(3b+1)+(3b+1)+(3c+1)+(3c+1)+(3a+1)大于等于2（根号下(3a+1)*(3b+1)+根号下(3b+1)*(3c+1)+根号下(3a+1)*(3c+1))
由a+b+c=1得：(3a+1)+(3b+1)+(3b+1)+(3c+1)+(3c+1)+(3a+1)=12大于等于2（根号下（3a+1)*(3b+1)+根号下(3b+1)*(3c+1)+根号下(3a+1)*(3c+1))