问题 问答题

证明:当x≥0时,

其中n为自然数。

答案

参考答案:

对f(x)求其导数,利用f’(x)的符号得到f(x)的单调区间及极大值,再对有关的不等式进行放大即可证得结果.

证 令

则有

f’(x)=(x-x2)sin2nx=x(1-x)sin2nx.

当0<x<1时,f’(x)>0,f(x)严格单调增加.

当x>1时,f’(x)<0(除去x=kπ(k=1,2,…)),f(x)严格单调递减.

因此x=1为f(x)的极大值点,且f(1)是x≥0时f(x)的最大值.

这说明,只要证明

则必有

成立.

事实上,当t≥0时,sint≤t.

所以当x≥0时,

阅读理解

阅读理解。

     Want to be an astronaut? Then "study hard, stay healthy and have perseverance (毅力)

      This is what Chinese astronauts Fei Junlong and Nie Haisheng said in a talk with some students.

      Students were interested in the astronauts' space lives. They asked questions like "Did you see the Great

Wall from space?" Fei and Nie said they didn't see the Great Wall. But they felt excited when they passed

over China, including Taiwan. The island "looked very beautiful, like a jade (翡翠)".

     Speaking of their lives beyond (在……之外) the earth, the two astronauts said cooperation (合作) with

each other was most important."We are like the left and right hands," they said.

      Young people wanted to know how the astronauts developed their careers (事业).

     "How did you decide to be astronauts when China's space industry (工业) is still at the beginning?" asked

a young woman. The astronauts said it was because of their love of flight and also the need of the country.

      They said their love of flight could be seen from their children's names. Nie's daughter is called "Nie

Tianxiang", which means "flying up high". Fei's son is called "Fei Di", which sounds like "Feidie", meaning

UFO.

     After the success of Shenzhou VI, China plans to put a man on the moon and build a space station in

10~15 years.

1. If you want to be astronauts, what should you do from now on?

A. We should change our names.

B. We should be interested in the astronauts' space life.

C. We should love the flight and our country.

D. We should study hard, stay healthy and have perseverance.

2. What's the Chinese meaning of the word "cooperation" in the passage? 

A. 友好

B. 竞争

C. 合作

D. 手术

3. In 10~15 years, China could put a man on ______ and build a space station.

A. Shenzhou VII

B. UFO

C. the moon

D. the sun

4. Which of the following is true?

A. Nie Haisheng saw the Great Wall from space.

B. Fei Junlong met UFO in space.

C. The astronauts couldn't see Taiwan from space.

D. Fei and Nie make a contribution to China's space industry

单项选择题