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问题 解答题
已知f（x）=ax-lnx，x∈（0，e]，g（x）=
lnx
x
，其中e是自然常数，a∈R．
（1）讨论a=1时，函数f（x）的单调性和极值；
（2）求证：在（1）的条件下，f（x）＞g（x）+
1
2
；
（3）是否存在实数a使f（x）的最小值是3？若存在，求出a的值；若不存在，说明理由．
答案

（1）因为f(x)=x-lnx，f′(x)=1-

1
x
=
x-1
x
，所以当0＜x＜1时，f'（x）＜0，此时函数f（x）单调递减．

  当1＜x≤e时，f'（x）＞0，此时函数f（x）单调递增．所以函数f（x）的极小值为f（1）=1．

（2）因为函数f（x）的极小值为1，即函数f（x）在（0，e]上的最小值为1．

又g′(x)=

1-lnx
x2
，所以当0＜x＜e时，=g'（x）＞0，此时g（x）单调递增．

所以g（x）的最大值为g（e）=

1
e
＜
1
2
，所以f(x)min-g(x)max＞
1
2
，所以在（1）的条件下，f（x）＞g（x）+
1
2
．

（3）假设存在实数a，使f（x）=ax-lnx，x∈（0，e]，有最小值3，则f′(x)=a-

1
x
=
ax-1
x
，

①当a≤0时，f'（x）＜0，f（x）在（0，e]上单调递减，f(x)min=f(e)=ae-1=3，a=

4
e
，（舍去），此时函数f（x）的最小值是不是3．

②当0＜

1
a
＜e时，f（x）在（0，
1
a
]上单调递减，f（x）在（
1
a
，e]上单调递增．

所以(x)min=f(

1
a
)=1+lna=3，a=e2，满足条件．

③当

1
a
≥e时，f（x）在（0，e]上单调递减，f(x)min=f(e)=ae-1=3，a=
4
e
，（舍去），此时函数f（x）的最小值是不是3．

综上可知存在实数a=e2，使f（x）的最小值是3．

填空题

根据要求写出相应的字母。(写出大小写形式)

1. 写出含元音音素 [ei] 的字母。 

                                                                               

2. 写出含元音音素 [i:] 的字母。 

                                                                               

3. 写出含元音音素 [ai] 的字母。 

                                                                               

4. 写出含有元音音素 [a:] 的字母。

                                                                               

5. 写出26字母中的元音字母。

                                                                               

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单项选择题

TOKY—Our kids, the Japanese government announced, have forgotten how to behave. They can’t be bothered with housework. If they see someone being wronged, they probably look the other way.
Few countries have placed more importance on being well-behaved in public than Japan. The simplest requests for directions often result in guided tours. Smiling shopkeepers are still the rule. Lost wallets usually make their way to their owners.
But according to recent surveys, all that may be going the way of the ancient hairdo. And Japan’s government has gone into something of a crisis mode.
A Japanese Education Ministry Survey formed late in 1999 and made public last month found that Japan moves behind other nations in teaching youngsters right from wrong.
It also reported that Japanese children are less helpful and do far less housework than their foreign peers in all classes. But they are better about taking dirty dishes to the kitchens after dinner.
In addition, Japanese kids are more likely to dry their hair and carry cell phones than American and Chinese kids, according to another survey, by a Tokyo-based tank.
Children in about 8 percent of public school classrooms are so disorderly that teachers cannot hold lessons, further recent reports show. children refuse to sit, to listen or to stop talking.
Older and middle-aged Japanese continue to have a solid sense of good manners and social justice, says Professor Yoshina Hirano from Shinshu University, who was appointed to direct the ministry’s survey.
Despite the knowledge of good manners among adults, the breakdown in manners may be spreading, he said.

It is implied in this passage that Japanese kids ______.

A. spending much time doing their homework
B. lead an advanced modern life
C. have their hair cut too often
D. often wash dishes after dinner
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