问题 解答题

已知函数f(x)=(ax﹣1)ex,a∈R

(1)当a=1时,求函数f(x)的极值.

(2)若函数f(x)在区间(0,1)上是单调增函数,求实数a的取值范围.

答案

解:(I)因为f'(x)=(ax+a﹣1)ex

所以当a=1时,f'(x)=xex

令f'(x)=0,则x=0,

所以f(x),f'(x)的变化情况如下表:

所以x=0时,f(x)取得极小值f(0)=﹣1.

(II)因为f'(x)=(ax+a﹣1)ex,函数f(x)在区间(0,1)上是单调增函数,

所以f'(x)≥0对x∈(0,1)恒成立.

又ex>0,所以只要ax+a﹣1≥0对x∈(0,1)恒成立,

解法一:设g(x)=ax+a﹣1,则要使ax+a﹣1≥0对x∈(0,1)恒成立,

只要成立,即,解得a≥1.

解法二:要使ax+a﹣1≥0对x∈(0,1)恒成立,

因为x>0,所以对x∈(0,1)恒成立,

因为函数在(0,1)上单调递减,

所以只要

阅读理解

阅读理解。

     A few days ago my wife sent me out to buy something. When I came home, I handed her a paper bag

and said, "Here are the apples you wanted." She looked in the bag, and then she looked at me. "I told you,"

she said slowly, "to buy some eggs."

     I felt worried about my absent-mindedness, so I went to see a doctor. He was a very kind man "I have

seen many people like you. It's nothing to worry about," he said "If you know you are absent-minded, you

are all right.But if you do not know you're absent-minded, you may be really in trouble."

     "Many famous people were absent-minded," the doctor told me. "Thomas Edison was standing in line

one day to pay his taxes. When he arrived at the window, he found that he had forgotten his own name.

He had to ask the man who was behind him to tell him who he was."

     I felt much better and got up to leave. "Thank you, doctor," I said. "How much do I have to pay you?"

     "Ten dollars for the check-up," the doctor said.

     "But doctor, I did not have a check-up!"

     For a moment the doctor looked puzzled. Then he said, "Oh, yes, it was the patient before you who had

a check-up. How absent-minded I am!"

1. The doctor took Edison's story for example ____.

A. to show absent-mindedness is common among people

B. because he liked telling stories

C. to make the author laugh

D. to make fun of Edison

2. The underlined word "absent-mindedness" in the second paragraph probably means ____.

A. a feeling of anger

B. a feeling of sadness

C. an emotion (情绪) of excitement

D. a trouble in which people often forget this or that

3. According to the passage, who was absent-minded?

A. Only the author.

B. The author's wife.

C. The doctor alone.

D. The author and the doctor.

填空题