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问题 解答题
已知函数f(x)=x+
a2
x
-3，g(x)=x+lnx，其中a＞0，F(x)=f(x)+g(x)．
（1）若x=
1
2
是函数，y=F（x）的极值点，求实数a的值；
（2）若函数y=F（x）（x∈（0，3]）的图象上任意一点处切线的斜率k≤
5
2
恒成立，求实数a的取值范围；
（3）若函数y=f（x）在[1，2]上有两个零点，求实数a的取值范围．
答案

F(x)=2x+

a2
x
+lnx-3，F′(x)=2-
a2
x2
+
1
x
（2分）

（1）F′(

1
2
)=4-4a2=0且a＞0，∴a=1（4分）

（2）F′(x)=2-

a2
x2
+
1
x
≤
5
2
对任意的x∈（0，3]恒成立（5分）

∴2a2≥-x2+2x对任意的x∈（0，3]恒成立，

∴2a2≥（-x2+2x）max，而当x=1时，-x2+2x=-（x-1）2+1取最大值为1，

∴2a2≥1，且a＞0，∴a≥

2
2
（8分）

（3）因为函数f(x)=x+

a2
x
-3在[1，2]上有两个零点，

所以方程a2=-x2+3x在x∈[1，2]上有两个不等实根（a＞0）（10分）

又因为函数y=-x2+3x=-(x-

3
2
)2+
9
4
在x∈[1，2]内的值域为[2，
9
4
]（12分）

由函数图象可得：2≤a2＜

9
4
，a＞0，所以：
2
≤a＜
3
2
，

即实数a的取值范围是[

2
，
3
2
)（14分）

解答题

化简：3（4x2-4x+1）-4（3x2+x-1）
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听力题

在短文的空格内填入适当的次,使其内容通顺. 每空格限填一词, 首字母已给 ): ( 14分)

At one time, making a film was an expensive activity only for a small group of people and film companies. Not only were the tools and machines expensive but the cost of film was also far greater than most people could afford. However, modem technology has c 小题1:  all that.

New technology has really opened up the world of film-making for film fans. Now a camera

does not cost m 小题2:  and most young people are certainly able to pay for it. So all you need is to get one for yourself. With that, you may then start your own film-making.

However, film-making is t 小题3:  work. You probably need many other people to help you.

For example, the p  小题4:   of actors and actresses is very important in your film. Of course, these actors and actresses may be your schoolmates, friends or family members. You don’t even need to pay them a cent while they are glad to be part of your job.

B 小题5:   , you need to make up a story. Can you write your own one? If you have good imagination, you can create an attractive story e  小题6: . Many world-famous film makers and producers are also good story-writers. If not, perhaps you need someone else to write it for you.

An excellent story is the key to making a s  小题7:   film.
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