Question

问一道高中数学题，如图。

Answer 1

(I)f(x)=x-alnx-(a-1)/x,a<2,x>0,
f'(x)=1-a/x+(a-1)/x^2
=(x-1)(x-a+1)/x^2,
max{0,a-1}<x<1时f'(x)<0,f(x)是减函数，其他，f(x)是增函数。
(III)g(v)=(1/2)v^2-e^v-ve^v,v∈[-2,0],
g'(v)=v-(2+v)e^v<0,
∴g(v)>=g(0)=-1,
存在u∈[e,e^2],使得对任v∈[-2,0],u-alnu-(a-1)/u<(1/2)v^2-e^v-ve^v,
<==>存在u∈[e,e^2],使得u-alnu-(a-1)/u<-1,
<==>存在u∈[e,e^2],使得u+1/u+1<a(lnu+1/u),
<==>存在u∈[e,e^2],使得a>(u^2+u+1)/(ulnu+1),记为h(u),①
h'(u)=[(2u+1)(ulnu+1)-(u^2+u+1)(lnu+1)]/(ulnu+1)^2
=(u-1)[(u+1)lnu-u]/(ulnu+1)^2>0,
∴h(u)>=h(e)=(e^2+e+1)/(e+1),
∴①<==>a>(e^2+e+1)/(e+1),为所求.

Answer 2

Answer 3

追答