已知a>0,b>0,且a+b=1,试用分析法证明不等式(a+1/a)(b+1/b)≥25/4

Question

已知a>0,b>0,且a+b=1,试用分析法证明不等式(a+1/a)(b+1/b)≥25/4
 

Answer 1

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
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