问题 解答题

已知函数f(x)=ax4lnx+bx4-c(x>0)在x=1处取得极值-3-c,其中a,b,c为常数.

(1)试确定a,b的值;

(2)讨论函数f(x)的单调区间;

(3)若对任意x>0,不等式f(x)≥-2c2恒成立,求c的取值范围.

答案

(1)由题意知f(1)=-3-c,因此b-c=-3-c,从而b=-3

又对f(x)求导得f′(x)=4ax3lnx+ax4

1
x
+4bx3=x3(4alnx+a+4b)

由题意f'(1)=0,因此a+4b=0,解得a=12

(2)由(I)知f'(x)=48x3lnx(x>0),令f'(x)=0,解得x=1

当0<x<1时,f'(x)<0,此时f(x)为减函数;

当x>1时,f'(x)>0,此时f(x)为增函数

因此f(x)的单调递减区间为(0,1),而f(x)的单调递增区间为(1,+∞)

(3)由(II)知,f(x)在x=1处取得极小值f(1)=-3-c,此极小值也是最小值,

要使f(x)≥-2c2(x>0)恒成立,只需-3-c≥-2c2

即2c2-c-3≥0,从而(2c-3)(c+1)≥0,解得c≥

3
2
或c≤-1

所以c的取值范围为(-∞,-1]∪[

3
2
,+∞)

阅读理解

阅读下面的短文,并根据短文后的要求答题。(请注意问题后的字数要求)

     Doctors say anger can be an extremely damaging, unless you learn how to deal with it. They warn

that anger can lead to heart disease, stomach problems, headaches, emotional problems and possibly

cancer.Anger is a normal emotion that we all feel from time to time, some people express anger openly

in a calm reasonable way. Others burst with anger, and scream and yell. But other people keep their

anger inside. They can not or will not express it. This is called ______________.

     For years many doctors thought that repressing anger was more dangerous to a person's health than

expressing it. They said that when a person is angry, the brain release the same hormones(荷尔蒙). They

speed the heart rate, raise blood pressure, or sugar into the blood, etc. in general the person feels excited

and ready to act.

     Some doctors say that both repressing and expressing anger can be dangerous. They believe that those who express anger violently may be likely to develop heart disease, and they believe that those who keep

 their anger inside may face a greater danger of high blood pressure.

     Doctors say the solution is learning how to deal with anger. They say the first step is to admit that you

are angry and to recognize the real cause of the anger, then decide if the cause is serious enough to get

angry about. If it is, they say, "Do not express your anger while angry. Wait until your anger has cooled

down and you are able to express yourself calmly and reasonably."

     Doctors say that a good way to deal with anger is to find humor in the situation that has made you

angry. They said that laughter is much healthier than anger.

1. What is the best title of the passage? (please answer within 10 words.)

______________________________________________________________

2. Which sentence in the passage can be replaced by the following one?

If you don't express your anger, you may have more chances of having high blood pressure.

_____________________________________________________________________________

3. Please fill in the blank with proper words or phrases to complete the sentence. (please answer within 10 words.)

_____________________________________________________________________________

4. What did you do when you became angry before? (please answer within 30 words.)

____________________________________________________________________________

5. Translate the underlined sentence into Chinese.

________________________________________________

单项选择题