问题 解答题
已知函数f
x
=ln|x|
x≠0
,函数g
x
=
1
f′
x
+af′
x
x≠0

(I)当x≠0时,求函数y=g
x
的表达式;
(Ⅱ)若a>0,且函数y=g
x
0,+∞
上的最小值是2,求a的值;
(Ⅲ)对于(Ⅱ)中所求的a值,若函数h(x)=
1
3
x3-
b+1
2a
x2+bx,x∈R
,恰有三个零点,求b的取值范围.
答案

(Ⅰ)∵f

x
=ln|x|,

∴当x>0时,f

x
=lnx; 当x<0时,f
x
=ln
-x

∴当x>0时,f′

x
=
1
x
; 当x<0时,f′
x
=
1
-x
-1
=
1
x

∴当x≠0时,函数y=g

x
=x+
a
x

(Ⅱ)∵由(1)知当x>0时,g

x
=x+
a
x

∴当a>0,x>0时,g

x
≥2
a
当且仅当x=
a
时取等号.

2

a
=2,得a=1,

(Ⅲ)h′(x)=x2-(b+1)x+b=(x-1)(x-b)

令h′(x)=0,得x=1或x=b.

(1)若b>1,则当0<x<1时,h′(x)>0,当1<x<b,时h′(x)<0,当x>b时,h′(x)>0;

(2)若b<1,且b≠0,则当0<x<b时,h′(x)>0,当b<x<1时,h′(x)<0,当x>1时,h′(x)>0.

所以函数h(x)有三个零点的充要条件为

f(1)>0
f(b)<0
f(1)<0
f(b)>0
解得b<
1
3
或b>3.

综合:b∈(-∞,0)∪(0,

1
3
)∪(3,+∞)

h(x)=

1
3
x3-
b+1
2
x2+bx=
1
6
x[2x2-3(b+1)x+6b]

所以,方程2x2-3(b+1)x+6b=0,有两个不等实根,且不含零根.

9(b+1)2-48b>0
b≠0
,解得:b∈(-∞,0)∪(0,
1
3
)∪(3,+∞)

完形填空

Ⅲ、完形填空(15分)

From Monday to Friday most people are busy working or studying,but in the evenings and weekends they are free and    31    themselves.Some watch television or go to the movies,others    32    sports.This is decided by their own     33    .

There are many different ways to spend our    34    time.Almost everyone has some kind of    35    :it may be something from collecting stamps to    36    model planes.Some hobbies are very    37   ,but others don’t cost anything at all. Some collections(收藏) are worth    38    of money, others are valuable(有价值的) only to their owners.

I know a man who has a coin collection worth several    39    dollars. A short time ago he bought a rare(罕见的)fifty-cent piece which    40    him 250 dollars. He was very happy about his collection and thought the price was all right. On the other hand, my youngest brother collects match boxes. He has almost 600 of them, but I wonder    41    they are worth any money. However,   42    my brother they are quite valuable.   43    makes him happier than to find a new match box for his collection.

That’s    44    a hobby means, I think. It is something we like to do in our free time just for the fun of it. The value in dollars is not important,   45   the pleasure it gives us is.

31. A. love           B. work         C. enjoy         D. play

32. A. look like       B. take care of   C. think about     D. take part in

33. A. lives          B. interests      C. jobs           D. things

34.A. working       B. free         C. own           D. whole

35 .A. hobby        B. thing       C. job             D. way

36 .A. make         B. making     C. makes         D. made

37.A. interesting     B. exciting     C. cheap        D. expensive

38 .A. a little        B. a few       C. a lot           D. a number

39 .A. thousand     B. thousands   C. thousands of   D. thousand of

40.A. paid          B. cost         C. took          D. spent

41 .A. that          B. if            C. what         D. why

42 .A. to            B. on          C. with          D. in

43 .A. Everything    B. Anything     C. Nothing       D. Something

44 .A. where        B. how         C. what         D. why

45.A. so            B. but          C. because      D. as

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