Question

已知f(x)=x/(x-a) (x≠a)

(1)若a=-2,试证f(x)在(-∞,-2)内单调递增 (2)若a＞0且f(x)在(1,+∞)内单调递减,求a的取值范围

Answer 1

1.若a=-2 假设-∞<x1<x2<-2 f(x1)-f(x2)=x1/(x1+2)-x2/(x2+2) =2(x1-x2)/[(x1+2)(x2+2)]<0 f(x)在(-∞,-2)内单调递增 2. 若a＞0且f(x)(1,+∞)内单调递减 假设1<x1<x2<∞ f(x1)-f(x2)>0 f(x1)-f(x2)=x1/(x1+a)-x2/(x2+a) =a(x2-x1)/(x2-a)(x1-a)>0 因为a(x2-x1)>0 (x2-a)(x1-a)>0 所以a<x1 即0<a<=1