Question

a，b是正实数且a+b=1 证明：ab+1/ab〉=4+1/4

很容易做出ab<=1/4 1/ab>=4 但不等式符号方向不同不能相加 请问大家接下来如何做啊？

还没学函数呢。。

Answer 1

a+b=1推出ab<=(a+b)^2/4=1/4
ab+1/ab=ab+1/(16ab)+15/(16ab)>=1/2+15/(4ab)>=1/2+15/4=17/4
当且仅当ab=1/(16ab)和ab=1/4即ab=1/4时有最小值17/4

Answer 2

a+b>=2√ab
=>
√ab<=1/2
=>
ab<=1/4

=>
ab+1/ab>=1/4+4
所以成立

Answer 3

a+b>=2√ab
=>
√ab<=1/2
=>
ab<=1/4

=>
ab+1/ab>=1/4+4
所以成立