杭州交通信息网长水痘吃什么比较好:高中数学题
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已知a、b是正实数，a+b=1,求证：(a+1/a)(b+1/b)≥25/4
(a+1/a)(b+1/b)
=ab+a/b+1/ab+b/a
=(a2b2+a2+1+b2)/ab
=[a2b2+(1-2ab)+1]/ab
=[(ab-1)2+1]/ab
(ab-1)2+1≥25/16,
ab≤1/4,
1/ab≥4
(a+1/a)(b+1/b)≥25/4.
证明：(a+1/a)(b+1/b)
=ab+a/b+1/ab+b/a
=(a2b2+a2+1+b2)/ab
=[a2b2+(1-2ab)+1]/ab
=[(ab-1)2+1]/ab
(ab-1)2+1≥25/16,
ab≤1/4,
1/ab≥4
(a+1/a)(b+1/b)≥25/4.