问题 填空题
关于函数f(x)=lg
x
x2+1
,有下列结论:
①函数f(x)的定义域是(0,+∞);
②函数f(x)是奇函数;
③函数f(x)的最小值为-lg2;
④当0<x<1时,函数f(x)是增函数;当x>1时,函数f(x)是减函数.
其中正确结论的序号是______.(写出所有你认为正确的结论的序号)
答案

①函数f(x)的定义域是(0,+∞),令

x
x2+1
>0,解得x>0,故定义域是(0,+∞),命题正确;

②函数f(x)是奇函数,由①知,定义域不关于原点对称,故不是奇函数,命题不正确;

③函数f(x)的最小值为-lg2,不正确,因为f(x)=lg

x
x2+1
=lg
1
x+
1
x
≤lg
1
2
=-lg2,最大值是-lg2,故命题不正确;

④当0<x<1时,函数f(x)是增函数;当x>1时,函数f(x)是减函数,命题正确,因为f′(x)=lg

1-x2
(x2+1)2
,令导数大于0,可解得0<x<1,令导数大于0,得x>1,故命题正确.

综上,①④正确

故答案为:①④

阅读理解

根据短文判断下列句子的正确(T)与错误(F)。

     At Harton College, there are many rules. Fifteen-year-old boy Bob Sanders often breaks these rules.

     The boys must return to school by 6 o'clock. One afternoon Bob walked to the town. He looked at the

shops and then went to the cinema. After the film, it was 8 o'clock. He walked back to Harton College as

fast as possible.

    When he arrived, the main entrance was locked. He looked up at the window of his dormitory. It was on

the third floor. He couldn't climb up the wall very easily. Then he saw another open window on the ground

floor. It was the window of the headmaster's study, and no one was there. He quickly climbed and jumped

into, the room. Just then he heard a noise. Then someone turned on a light in the corridor (走廊). Bob had

to hide under the sofa. One minute later, Mr. Mason, the headmaster, came in. He turned on the light and

then he sat down on the sofa and began to read a book.

     Bob lay under the sofa as quietly as possible. He looked at the headmaster's shoes and socks for nearly

an hour.

     Finally, the headmaster closed his book and walked towards the door.

     "Thank Heavens, he didn't find me under the sofa!" thought Bob.

     Then Mr. Mason stopped and spoke towards the sofa. "Would you turn off the light when you leave?"

He said and left the study.

( )1. Bob is 16 years old.

( )2. The students must return to school by 6 0'clock.

( )3. The school gate was locked when he returned to school.

( )4. Bob hid himself under the sofa for over one hour.

( )5. Mr. Mason didn't find Bob under the sofa.

选择题