问题 解答题
设函数f(x)=lnx+aln(2-x).
(Ⅰ)求函数f(x)的定义域及其导数f'(x);
(Ⅱ)当a≥-1时,求函数f(x)的单调区间;
(Ⅲ)当a=1时,令g(x)=f(x)+mx(m>0),若g(x)在(0,1]上的最大值为
1
2
,求实数m的值.
答案

(Ⅰ)由

x>0
2-x>0
得0<x<2,即函数的定义域为(0,2);

f′(x)=

1
x
-
a
2-x

(Ⅱ)当a≥-1时,f′(x)=

1
x
-
a
2-x
=
2-(a+1)x
x(2-x)

当a=-1时,f′(x)=

2
x(2-x)
,所以在区间(0,2)上,f'(x)>0,

故函数f(x)的单调递增区间是(0,2);

当a>-1时,令f′(x)=

2-(a+1)x
x(2-x)
=0,解得x=
2
a+1

①当

2
a+1
≥2时,即-1<a≤0时,在区间(0,2)上,f'(x)>0,

故函数f(x)的单调递增区间是(0,2);

②当0<

2
a+1
<2时,即a>0时,在区间(0,
2
a+1
)
上,f'(x)>0,

在区间(

2
a+1
,2)上,f'(x)<0,故函数f(x)的单调递增区间是(0,
2
a+1
)
,单调递减区间是(
2
a+1
,2)

(Ⅲ)当x∈(0,1]且m>0时,g′(x)=

1
x
-
1
2-x
+m=
2(1-x)
x(2-x)
+m>0,

即函数在区间(0,1]上是增函数,故函数g(x)在(0,1]上的最大值为g(1),

所以g(1)=m=

1
2
,即m=
1
2

单项选择题 B1型题
阅读理解

   In the future your automobile will run on water instead of gas! You will be able to buy a supercomputer that fits in your pocket! You might even drive a flying car!

For each prediction that has come true today, several others have missed by a mile. Many of these predictions didn’t consider how people would want to use the technology, or if people really needed it in their lives or not. Let’s look at some predictions from the not-too-distant past.

Robot Helpers

Where’s the robot in my kitchen? Nowhere, of course. And he’s probably not coming anytime soon. Robots do exist today, but mostly in factories and other manufacturing environments.

Back in the 1950s, however, people said that by now personal robots would be in most people’s homes.

So why hasn’t it happened?  Probably because robots are still too expensive and clumsy. And maybe the idea of robots cooking our dinners and washing our clothes is just too weird. At home we seem to be doing fine without them.

Telephones of Tomorrow

In 1964 an American company introduced the video telephone. They said by the year 2000 most people would have a video phone in their homes. But of course the idea hasn’t caught on yet.

Why? The technology worked fine, but it over—looked something obvious: people’s desire for privacy. Would you want to have a video phone conversation with someone after you just step out of the shower?  Probably not---it could be embarrassing! Just because a technology available doesn’t always mean people will want to use it.

And finally, how about that crazy prediction of the flying car? It’s not so crazy anymore! But a flying car remains one of the most fascinating technology ideas to capture our imagination. Keep watching the news, or perhaps the sky outside your window, to see what the future will bring.

64. The whole passage is mainly about ________________.

A. predictions that have come true.

B. predictions that haven’t come true.

C. why predictions don’t come true easily.

D. what technology will bring about.

65. The author of this passage won’t believe that _________________.

A. predictions needn’t consider people’s practical use of technology.

B. the future isn’t always easy to guess.

C. not all past predictions have come true.

D. many of the high—tech things our parents thought we’d be using by now simply never appeared.

66. The underlined word “weird” probably means __________.

A. wonderful   B. stupid    C. practical    D. strange

67. What does the author think of the flying car?

A. It is too difficult to imagine.

B. It is too crazy an idea.

C. It is likely to be made.

D. It is often reported in the news.