Question

己知函数f(x)=alnx+bx(a,b∈r)在x=1/2处取得极值且曲线y=f(x)在(1,f

x))处的切线与直线x-y+1=0垂直
(1)求实数a、b的值
(2)若对任意x∈[1,+∞)，不等式f(x)≤(m-2)x-m/x恒成立，求实数m的取值范围

Answer 1

(1)f'(x)=a/x+b
f'(1/2)=2a+b=0
f'(1)=a+b=-1
a=1 b=-2
(2)F(x)=lnx-2x-(m-2)x+m/x F(1)=0
1/x-2-m+2-m/x^2<0 x>=1
1/x-m-m/x^2<0 x>=1
△=1-4m^2<0
m^2>1/4
m>1/2 或 m<-1/2