a>1b>0 a+b=2 1/(a-1)+2/b等于

题目

a>1b>0 a+b=2 1/(a-1)+2/b等于

答案

b=2-a,
1/(a-1)+2/b=1/(a-1)+2/(2-a),记为f(a),1f'(a)=-1/(a-1)^2+2/(2-a)^2
=[-(2-a)^2+2(a-1)^2]/[(a-1)(2-a)]^2
=(a-√2)(a+√2)/[(a-1)(2-a)]^2,
1√2时f'(a)>0,
∴f(a)|min=f(√2)=1/(√2-1)+2/(2-√2)=3+2√2.