问题 解答题

已知函数f(x)=x2-ax+4+2lnx

(I)当a=5时,求f(x)的单调递减函数;

(Ⅱ)设直线l是曲线y=f(x)的切线,若l的斜率存在最小值-2,求a的值,并求取得最小斜率时切线l的方程;

(Ⅲ)若f(x)分别在x1、x2(x1≠x2)处取得极值,求证:f(x1)+f(x2)<2.

答案

(I)因为函数的定义域为{x|x>0},

当a=5时,f(x)=x2-5x+4+2lnx,f′(x)=2x-5+

2
x
=
2x2-5x+2
x
=
2(x-
1
2
)(x-2)
x

所以由f'(x)<0,解得

1
2
<x<2,

即函数的单调递减区间为(

1
2
,2).

(Ⅱ)因为x>0,所以f′(x)=2x+

2
x
-a≥2
4
-a=4-a,

当且仅当x=1时取等号.因为直线l的斜率存在最小值-2,

所以4-a=-2,即a=6.

当l取得最小斜率时,因为f(-1)=-1,即切点为(1,-1).

从而切线方程l:y+1=-2(x-1),即:2x+y-1=0.

(Ⅲ)f′(x)=2x+

2
x
-a=
2x2-ax+2
x

因为f(x)分别在x1、x2(x1≠x2)处取得极值,

所以x1、x2(x1≠x2)是方程

2x2-ax+2
x
=0,

即2x2-ax+2=0的两个不等正根.

则△=a2-16>0解得a2>16,且x1+x2=

a
2
x1x2=1.

从而f(x1)+f(x2)=

x21
+
x22
-a(x1+x2)+8+2ln⁡(x1x2)

=(x1+x2)2-2x1x2-a(x1+x2)+8+2ln⁡(x1x2)

=(

a
2
)2-2×1-a×
a
2
+8+2ln1=-
a2
4
+6,

因为a2>16,所以-

a2
4
+6<2.

即不等式f(x1)+f(x2)<2成立.

阅读理解

阅读理解      

       How to greet people from a different country? Most people will say " Hello " or " Nice to meet

you ".  That's right. But greeting is never just that. To be both polite and friendly , you should know

something more.

        In most western countries, people usually shake hands when they meet. The British  only  say

" hello" when they see friends. That's why French people think the British are unfriendly and impolite.

They kiss friends on both cheeks(面颊) each time they meet and leave. How-ever, to Americans,

kiss is very strange between two male friends.

       When you come to Polynesia, you can take people's hands to touch your face. That's a way to

show friend-ship. In Canada, people rub noses when they meet friends. In Tibet, China, you should

bow and stick your tongue out. It shows you have no bad thoughts because you cannot curse(诅咒)

people when your tongue's out.

       Some of the ways above to greet people might be strange for you, but do remember: when in

Rome, do as the Romans do.

1. What can we infer from the first paragraph?   

A. The ways of greet people in different countries are different.

B. Most people like  to  say " Hello" than" Nice  to  meet you " .

C. "Hello" and'- Nice to meet you" are easy for people to remember.

D. There are some more greetings words except " Hello " or" Nice  to  meet  you " .

2. The underlined word" they" refers to ________ . 

A. Americans          

B. Chinese

C. British              

D. French

3. Canadians  ________  to show friendship when they meet friends.

A. take people's hands to touch their faces

B. rub noses 

C. stick tongues out

D. bow

4. In Tibet, China, people stick their tongues out to show ________.

A. they are unhappy

B. they have no bad thoughts

C. they want to curse others

D. they are excited

5. What's the best title for the passage? 

A. Greeting people in different countries

B. The differences  between" Hello" and " Nice to meet you "

C. Different customs in different countries

D. The polite and friendly way to greet people

单项选择题