若 a,b是两个正数,且…………且（a-1）/b +（b-1）/a +1=0,求a、b的取值范围

若 a,b是两个正数,且…………且（a-1）/b +（b-1）/a +1=0,求a、b的取值范围
若 a,b是两个正数,且…………
且（a-1）/b +（b-1）/a +1=0,求a、b的取值范围

若 a,b是两个正数,且…………且（a-1）/b +（b-1）/a +1=0,求a、b的取值范围
(a-1)/b+(b-1)/a+1=0
两边同乘ab得：
a^2-a+b^2-b+ab=0
(a+b)^2-2ab-(a+b)+ab=0
ab=(a+b)^2-(a+b)
又：
(a-b)^2≥0
a^2+b^2≥2ab
(a+b)^2≥4ab
ab≤1/4(a+b)^2
所以：ab=(a+b)^2-(a+b)≤1/4(a+b)^2
3/4(a+b)^2-(a+b)≤0
(a+b)[3/4(a+b)-1]≤0
∵a＞0,b＞0
∴(a+b)＞0
∴3/4(a+b)-1≤0
∴(a+b)≤4/3
∴0＜a+b≤4/3,

因为（a-1）/b +（b-1）/a +1=0
(a-1)/b+(b-1)/a+1=[a(a-1)+b(b-1)+ab]/ab=0
故有a²+b²-(a+b)+ab=0
(a+b)²-(a+b)-ab=0.......................(1)
(a+b)²-(a+b)=ab≤(a+b)²/4

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因为（a-1）/b +（b-1）/a +1=0
(a-1)/b+(b-1)/a+1=[a(a-1)+b(b-1)+ab]/ab=0
故有a²+b²-(a+b)+ab=0
(a+b)²-(a+b)-ab=0.......................(1)
(a+b)²-(a+b)=ab≤(a+b)²/4
故有1-1/(a+b)≤1/4，1/(a+b)≥1-1/4=3/4，故a+b≤4/3
又由（1）得 (a+b)(a+b-1)=ab
由于a，b都是正数，故必有ab>0，a+b>0，a+b-1>0，由此得a+b>1
故a+b的取值范围为：1

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