Question

高二数学18

高二数学18第18

Answer 1

18、
(1) f(x)=2ax-b/x+lnx
f'(x)=2a+b/x^2+1/x
f'(1)=0
2a+b+1=0..........(1)
f'(1/2)=0
2a+b/(1/2)^2+1/(1/2)=0
2a+4b+2=0
a+2b+1=0........(2)
联解(1)(2)：a=-1/3，b=-1/3
(2) f(x)=-2/3x+1/(3x)+lnx
f'(x)=-2/3-1/(3x^2)+1/x
=(-2x^2-1+3x)/(3x^2)
=-(x-1)(2x-1)/(3x^2)
f'(x)>0
-(x-1)(2x-1)/(3x^2)>0
(x-1)(2x-1)<0
1/2<x<1
单调递增区间：(1/2,1)
单调递减区间：(0,1/2)U(1,+∞)
极小值：f(1/2)=-2/3(1/2)+2/3+ln(1/2)
=-1/3+2/3-ln2
=1/3-ln2
≈-0.36
f(2)=-2/3×2+1/(3×2)+ln2
=-4/3+1/6+ln2
≈-0.474
最小值：f(x)min=f(2)≈-0.474
f(x0)-c<=0

-2/3x0+1/(3x0)+lnx0-c<=0
c>=-2/3x0+1/(3x0)+lnx0
x0=2时，c最小值：cmin=f(2)=-0.474