Question

已知a,b属于正实数,且a+b=1,求y=(a+a/1)(b+b/1)的最小值

Answer 1

(a+1/a)(b+1/b)
=ab+b/a+a/b+1/(ab)
=(a^2b^2+b^2+a^2+1)/(ab)
=[a^2b^2+(a+b)^2-2ab+1]/(ab)
=[a^2b^2+1-2ab+1]/(ab)
=[(ab-1)^2+1]/(ab)

1=a+b>=2√(ab)
所以√(ab)<=1/2
ab<=1/4
a>0,b>0
所以0<ab<=1/4
所以ab-1<=-3/4
所以(ab-1)^2>=(-3/4)^2=9/16
所以(ab-1)^2+1>=25/16

因为0<ab<=1/4
所以1/ab>=4

所以[(ab-1)^2+1]/(ab)>=4*25/16=25/4
即(a+1/a)(b+1/b)>=25/4
当且仅当a=b=5/2时取等号，故(a+1/a)(b+1/b)最小值25/4

Answer 2

当a=b=1/2时 最小吧 4

Answer 3

(a+1/a)(b+1/b)
=ab+b/a+a/b+1/(ab)
=(a^2b^2+b^2+a^2+1)/(ab)
=[a^2b^2+(a+b)^2-2ab+1]/(ab)
=[a^2b^2+1-2ab+1]/(ab)
=[(ab-1)^2+1]/(ab)
1=a+b>=2√(ab)
所以√(ab)<=1/2
ab<=1/4
a>0,b>0
所以0<ab<=1/4
所以ab-1<=-3/4
所以(ab-1)^2>=(-3/4)^2=9/16
所以(ab-1)^2+1>=25/16
因为0<ab<=1/4
所以1/ab>=4
所以[(ab-1)^2+1]/(ab)>=4*25/16=25/4
即(a+1/a)(b+1/b)>=25/4
当且仅当a=b=5/2时取等号，故(a+1/a)(b+1/b)最小值25/4

Answer 4

(a+1/a)(b+1/b)
=ab+b/a+a/b+1/(ab)
=(a^2b^2+b^2+a^2+1)/(ab)
=[a^2b^2+(a+b)^2-2ab+1]/(ab)
=[a^2b^2+1-2ab+1]/(ab)
=a^2b^2/ab-2ab/ab+2/ab=ab+2/ab-21=a+b>=2√(ab)所以√(ab)

Answer 5

均值定理