已知x>0,y+0,x+y=1则(1+1⼀x)(1+1⼀y)的最小值是?

2024-12-04 16:37:03
推荐回答(2个)
回答1:

(1+1/x)(1+1/y)
=1+1/x+1/y+1/xy
=1+(x+y)/x+(x+y)/y+(x+y)²/xy
=1+1+y/x+1+x/y+x/y+2+y/x
=5+2(y/x+x/y)
>=5+2×2√y/x*x/y
=5+4
=9
最小值=9

回答2:

4xy<=(x+y)^2=1, xy<=1/4
(1+1/x)(1+1/y)=(x+1)(y+1)/xy=(2+xy)/xy=1+2/xy>=1+8=9