问题 解答题
已知函数f(x)=px-
p
x
-2lnx

(Ⅰ)若p=2,求曲线f(x)在点(1,f(1))处的切线方程;
(Ⅱ)若函数f(x)在其定义域内为增函数,求正实数p的取值范围;
(Ⅲ)设函数g(x)=
2e
x
,若在[1,e]上至少存在一点x0,使得f(x0)>g(x0)成立,求实数p的取值范围.
答案

(I)当p=2时,函数f(x)=2x-

2
x
-2lnx,f(1)=2-2-2ln1=0.f′(x)=2+
2
x2
-
2
x

曲线f(x)在点(1,f(1))处的切线的斜率为f'(1)=2+2-2=2.

从而曲线f(x)在点(1,f(1))处的切线方程为y-0=2(x-1)

即y=2x-2.

(II)f′(x)=p+

p
x2
-
2
x
=
px2-2x+p
x2

令h(x)=px2-2x+p,

要使f(x)在定义域(0,+∞)内是增函数,只需h(x)≥0在(0,+∞)内恒成立.

由题意p>0,h(x)=px2-2x+p的图象为开口向上的抛物线,对称轴方程为x=

1
p
∈(0,+∞),

h(x)min=p-

1
p
,只需p-
1
p
≥0

即p≥1时,h(x)≥0,f'(x)≥0

∴f(x)在(0,+∞)内为增函数,正实数p的取值范围是[1,+∞).

(III)∵g(x)=

2e
x
在[1,e]上是减函数,

∴x=e时,g(x)min=2;x=1时,g(x)max=2e,

即g(x)∈[2,2e],

1当p<02时,h(x)=px2-2x+p3,其图象为开口向下的抛物线,对称轴x=

1
p
4在y5轴的左侧,且h(0)<0,

所以f(x)在x∈[1,e]9内是减函数.

当p=0时,h(x)=-2x,因为x∈[1,e],所以h(x)<0,

f′(x)=-

2x
x2
<0,此时,f(x)在x∈[1,e]内是减函数.

∴当p≤0时,f(x)在[1,e]上单调递减⇒f(x)max=f(1)=0<2,不合题意; (

当0<p<1时,由x∈[1,e]⇒x-

1
x
≥012,所以f(x)=p(x-
1
x
)-2lnx≤x-
1
x
-2lnx

又由(2)知当p=1时,f(x)在[1,e]上是增函数,

x-

1
x
-2lnx≤e-
1
e
-2lne=e-
1
e
-2<2,不合题意;

14当p≥115时,由(2)知f(x)16在[1,e]17上是增函数,f(1)=0<218,又g(x)19在[1,e]20上是减函数,

故只需f(x)max>g(x)min,x∈[1,e],而f(x)max=f(e)=p(e-

1
e
)-2lne,g(x)min=2,即p(e-
1
e
)-2lne>2
,解得p>
4e
e2-1

综上所述,实数p的取值范围是(

4e
e2-1
,+∞).

阅读理解

Botany (植物学) , the study of plants, occupies a particular position in the history of human knowledge.For many thousands of years it was the one field of awareness about which humans had anything more than the vaguest (模糊的)) of insights.It is impossible to know today just what our Stone Age ancestors knew about plants, but from what we can observe of pre-industrial societies that still exist, a detailed learning of plants and their properties must be extremely ancient.This is logical.Plants are the basis of the food pyramid for all living things even for other plants.They have always been enormously important to the welfare of people not only for food but also for clothing, tools, medicines, shelter, and a great many other purposes.Tribes living today in the jungle of the Amazon recognize hundreds of plants and know many properties of each.To them, botany has no name and is probably not even recognized as a special branch of "knowledge" at all.

Unfortunately, the more industrialized we become, the farther away we move from direct contact with plants, and the less distinct our knowledge grows.Yet everyone comes unconsciously on an amazing amount of botanical knowledge, and few people will fail to recognize a rose or an apple.When nor Neolithic ancestors, living in the Middle East about 10,000 years ago, discovered that certain grasses could be harvested and their seeds planted for richer fields the next season—the first, great step in a new association of plants and humans was taken.Grains were discovered and from them flowed the wonder of agriculture: cultivated crops.From then on, humans would increasingly take their living from the controlled production of a few plants, rather than getting a little here and a little there from many varieties that grew wild, and the accumulated knowledge of tens of thousands of years of experience with plants in the wild would begin to fade away.

小题1:What does the writer mean when he says "This is logical" in the first paragraph?

A.Human knowledge of plants is well organized and very detailed.

B.It is not surprising that early humans had a detailed knowledge of plants.

C.There is no clear way to determine the extent of our ancestor's knowledge of plant.

D.It is reasonable to assume our ancestors behave much like people in pre-industrial societies.小题2:According to the passage, general knowledge of botany begun to fade away because ____.

A.people no longer value plants as a useful resource

B.direct contact with a variety of plants has decreased

C.botany is not recognized as a special branch of science

D.research is unable to keep up with the increasing number of plants小题3:We learn that the first great step toward the practice of agriculture is____.

A.the changing diets of early humans

B.the development of a system of names for plants

C.the invention of agricultural implements (工具) and machinery

D.the discovery of certain grasses that could be harvested and replanted小题4:Human would depend on the controlled production of a few plants for living with

A.the knowledge of plants

B.the discovery of certain grasses

C.the development of machinery

D.the appearance of agriculture

单项选择题