问题 解答题
若f(x0)是函数f(x)在点x0附近的某个局部范围内的最大(小)值,则称f(x0)是函数f(x)的一个极值,x0为极值点.已知a∈R,函数f(x)=lnx-a(x-1).
(Ⅰ)若a=
1
e-1
,求函数y=|f(x)|的极值点;
(Ⅱ)若不等式f(x)≤-
ax2
e2
+
(1+2a-ea)x
e
恒成立,求a的取值范围.
(e为自然对数的底数)
答案

(Ⅰ)若a=

1
e-1
,则f(x)=lnx-
x-1
e-1
f′(x)=
1
x
-
1
e-1

当x∈(0,e-1)时,f'(x)>0,f(x)单调递增;

当x∈(e-1,+∞)时,f'(x)<0,f(x)单调递减.…(2分)

又因为f(1)=0,f(e)=0,所以

当x∈(0,1)时,f(x)<0;当x∈(1,e-1)时,f(x)>0;

当x∈(e-1,e)时,f(x)>0;当x∈(e,+∞)时,f(x)<0.…(4分)

故y=|f(x)|的极小值点为1和e,极大值点为e-1.…(6分)

(Ⅱ)不等式f(x)≤-

ax2
e2
+
(1+2a-ea)x
e

整理为lnx+

ax2
e2
-
(1+2a)x
e
+a≤0.…(*)

g(x)=lnx+

ax2
e2
-
(1+2a)x
e
+a,

g′(x)=

1
x
+
2ax
e2
-
1+2a
e
(x>0)=
2ax2-(1+2a)ex+e2
e2x
=
(x-e)(2ax-e)
e2x
.…(8分)

①当a≤0时,2ax-e<0,又x>0,所以,

当x∈(0,e)时,g'(x)>0,g(x)递增;

当x∈(e,+∞)时,g'(x)<0,g(x)递减.

从而g(x)max=g(e)=0.

故,g(x)≤0恒成立.…(11分)

②当a>0时,g′(x)=

(x-e)(2ax-e)
e2x
=(x-e)(
2a
e2
-
1
ex
)

2a
e2
-
1
ex
=
a
e2
,解得x1=
e
a
,则当x>x1时,
2a
e2
-
1
ex
a
e2

再令(x-e)

a
e2
=1,解得x2=
e2
a
+e
,则当x>x2时,(x-e)
a
e2
>1

取x0=max(x1,x2),则当x>x0时,g'(x)>1.

所以,当x∈(x0,+∞)时,g(x)-g(x0)>x-x0,即g(x)>x-x0+g(x0).

这与“g(x)≤0恒成立”矛盾.

综上所述,a≤0.…(14分)

阅读理解

阅读理解。

     Visitors to London often eat in restaurants. The owners and workers in them are all from other

countries. The visitors say in these restaurants they don't feel they are in England. Most English people

eat at home as much as they can. Sometimes they themselves will think they are in another country when

they are in restaurants. When an Englishman goes out of a restaurant, he may find that he doesn't

understand why everything is written in French and Italian.

      Most English people think it is better to eat at home, because it is cheaper. They don't want to spend

much money on food and they like cooking at home. They like fast food. But for Christmas, they will

spend two or three weeks to get ready for it, because they want to have good food.

l. Visitors to London often eat in restaurants because __________.

A. they don't like the food at home

B. their homes are not in London

C. it's cheaper

D. restaurants are beautiful places

2. When English people eat in restaurants, they sometimes think                     .

A. they are in another country

B. they are in the country

C. they are themselves

D. they love their country

3. Some English people don't know                     .

A. why everything comes from French and Italian

B. why people write everything in French and Italian

C. everything in restaurants

D. everyone in restaurants

4. English people eat at home because                     .

A. it is fast

B. the food is expensive at home

C. it is sometimes very cheap

D. the food is very cheap at home

5.                    English people will spend much time for Christmas.

A. In order to get more food

B. In order to have much food

C. In order to get good food

D. In order to have enough food

问答题