问题 填空题

已知定义在区间(0,+∞)的非负函数f(x)的导数为f'(x),其满足xf'(x)+f(x)<0,则在0<a<b时,下列结论一定正确的是______.

(1)af'(a)<bf'(b)(2)af(a)>bf(b)(3)bf(a)>af(b)(4)bf'(a)>af'(b)

答案

设g(x)=xf(x)所以g′(x)=xf'(x)+f(x)<0,所以g(x)在在区间(0,+∞)是减函数.因为0<a<b所以af(a)>bf(b).故(2)正确.

因为f(x)是定义在区间(0,+∞)的非负函数,并且xf'(x)+f(x)<0所以f′(x)<0.所以f(x)是定义在区间(0,+∞)的减函数.因为0<a<b所以f(a)>f(b),所以bf(a)>af(b).故(3)正确.

对于(1)(4)我们只知道0<a<b且f'(a)<0,f'(b)<0而并不能判断f'(a)与f'(b)的大小,所以af'(a)与bf'(b)、bf'(a)与af'(b)的大小不能判断.故(1)(4)不正确.

故答案为(2)(3).

阅读理解

Once upon a time there was a crook(骗子)who only thought about how to get money. What he hated most was when people were polite to each other, saying things like “please”, “thank you”, and “don’t mention it”.

The crook thought all those kinds of words were a waste, so he spent a lot of time inventing a machine which could steal words. With this machine, he planned to steal “please”, “thank you”, “don’t mention it”, and similar words people used to be polite. After he stole these words, he intended to take them apart and sell the letters to book publishers. After he started up his machine, people would open their mouths, intending to say kind and polite things, but nothing came out. All those words ended up inside the big machine.

The crook was happy with his success, but he didn’t count on (料想到)a couple of very special little girls. Those girls were deaf, and had to communicate with sign language. Because the machine couldn’t steal gestures, these girls continued being kind and polite. Soon they realized what had been happening to everyone else, and they found out about the crook and his plan. The girls followed him and found the machine busy separating all the words into letters. They approached the machine and started being polite to each other.

Try as it might, the machine couldn’t steal those words, and it started to suffer a power overload(超载). Finally, it exploded, sending all the letters it had gathered flying into the sky. These letters started coming down, like rain, and most ended up in the sea. After that, everyone could be polite again.

Seeing all those letters fall into the sea, those little girls had an idea. A little while later they opened a factory, making alphabetic spaghetti soup!

小题1:What the crook hated most was ______.

A.his machine

B.money

C.language

D.politeness小题2:The crook spent a lot of time inventing a machine to ______.

A.publish books

B.steal words

C.write letters

D.make alphabetic soup小题3:Why did the machine have no effect on the two girls?

A.Because they were kind and polite.

B.Because they were very clever.

C.Because the machine couldn’t steal gestures.

D.Because there was something wrong with the machine.

问答题