ab≤1/4
(a+1/a)(b+1/b)
=(a^2+1)/a*(b^2+1)/b
=[a^2b^2+(1-2ab)+1]/ab
=[(ab-1)^2+1]/ab
(ab-1)^2+1≥25/16
0
取等号时a=b=1/2
已知a＞0,b＞0,a＋b＝1,求证（a＋1／a）（b＋1／b）大于等于25／4
已知a＞0,b＞0,a＋b＝1,求证（a＋1／a）（b＋1／b）大于等于25／4
其他人气：585 ℃时间：2019-11-04 14:26:58
优质解答
a+b≥2√ab
ab≤1/4
(a+1/a)(b+1/b)
=(a^2+1)/a*(b^2+1)/b
=[a^2b^2+(1-2ab)+1]/ab
=[(ab-1)^2+1]/ab
(ab-1)^2+1≥25/16
0
取等号时a=b=1/2
ab≤1/4
(a+1/a)(b+1/b)
=(a^2+1)/a*(b^2+1)/b
=[a^2b^2+(1-2ab)+1]/ab
=[(ab-1)^2+1]/ab
(ab-1)^2+1≥25/16
0
取等号时a=b=1/2
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