已知函数f(x)=alnx-1/x,a∈R
(1)若曲线y=f(x)在点(1,f(1))处的切线与直线x+2y=0垂直,求a的值
f’(x)=a/x+1/x^2
f’(1)=a+1
x+2y=0斜率k=-1/2
切线斜率=2
a+1=2
a=1
(2)求函数f(x)的单调区间
f’(x)=a/x+1/x^2
令f’(x)=0
a/x+1/x^2=0
x=-1/a
f’’(x)=-a/x^2-2/x^3
f’’(-1/a)=-a/(1/a^2)+2/(1/a^3)
=-a^3+2a^3=a^3
a>0 f’’(-1/a)>0
f(-1/a)为极小值
x-1/a f(x)增
a