早教吧作业答案频道 -->数学-->
a>0 b>0 a+b=1 试用分析法证明(a+1/a)(b+1/b)≥25/4RT
题目详情
a>0 b>0 a+b=1 试用分析法证明(a+1/a)(b+1/b)≥25/4
RT
RT
▼优质解答
答案和解析
由均值不等式
a+b≥2√ab
ab≤1/4
证法一
(a+1/a)(b+1/b)
=(a^2+1)/a*(b^2+1)/b
=(a^2b^2+a^2+1+b^2)/ab
=[a^2b^2+(a+b)^2-2ab+1]/ab
=[a^2b^2+(1-2ab)+1]/ab
=[(ab-1)^2+1]/ab
(ab-1)^2+1≥25/16
0
a+b≥2√ab
ab≤1/4
证法一
(a+1/a)(b+1/b)
=(a^2+1)/a*(b^2+1)/b
=(a^2b^2+a^2+1+b^2)/ab
=[a^2b^2+(a+b)^2-2ab+1]/ab
=[a^2b^2+(1-2ab)+1]/ab
=[(ab-1)^2+1]/ab
(ab-1)^2+1≥25/16
0
看了 a>0 b>0 a+b=1 ...的网友还看了以下：