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高三数学周周练(六)


高三数学周周练(六)

1.函数 y ? log 2 x ? 2 的定义域为_____________。[4,??)
2.已知集合 A ? ?x | x ? a?, B ? ?x |1? x ? 2? ,且 A ? (CRB) ? R ,则实数 a 范围为: a ? 2
3.设函数 f0 (x) ?| x |, f1 (x) ?| f0 (x) ?1 |, f2 (x) ?| f1(x) ? 2 | ,则函数 y ? f2 (x) 的图象与 x 轴所围
成的图形中封闭部分的面积为 7 .
4.二次函数 y ? x2 ? ax ?1 在区间[0,3] 上有最小值 ?2, 则实数 a 的值为 2 .

5.函数

y

?

2

cos x cos x ?1

的值域是____

? ??

??,

1 3

? ??

?1, ??? ______.

6.设函数 f (x) ? n ?1, x ?[n, n ?1), n ? N ,则满足方程 f (x) ? log2 x 的根的个数是 3 .

7.设

f

(x)

?

?21?x ?

?

a,

x

?

0

,若

f

(x)

?

x

有且只有两个实数根,则实数 a

的取值范围是[3, 4)

.

? f (x ?1), x ? 0

8.给出四个命题:①函数 y ? a|x|与y ? loga | x | (a ? 0, a ? 1) 的图象关于直线 y ? x 对称;②函数

y ? a|x|与y ? ( 1 )|x| a

(a ? 0, a ? 1)

的图象关于

y

轴对称;③函数

y ? loga | x |



y ? log1 | x | (a ? 0, a ? 1) 的图象关于 x 轴对称,其中正确的命题是 ③ .

a

9.已知 f (x) ? (x ? 1)? | x ? 1 |,若关于 x 的方程 f (x) ? x ? m 有三个不同的实数解,则实数 m 的

取值范围是 (?1, 5) . 4

10.函数 f

(x)=

?lg | ??1(x

x ?

?2 2)

|

(

x

?

2)

若关于 x 的方程[f (x)] 2 +b·f (x)+C=0,恰有 3 个不

同的实数解 x1、x2、x3,则 f (x1+x2+x3)等于

。lg4

11.(理)直线 y ? a 与函数 f (x) ? x3 ? 3x 的图像有相异的三个公共点,则 a 的取值范围是_▲(-2,

2)
(文)一个等比数列的公比 q 满足 0 ? q ? 1,且它的第 4 项与第 8 项之和为 17 ,而第 5 项与第 7 8

项之积等于 1 4

.若它的前 n

项的和 Sn

?

31,则满足条件的自然数 n

的最小值为

6.

12.已知二次函数

f

(x)

?

ax2

?

2x

?

c(x

?

R)

的值域为 [0,

??)

,则

a

?1

?

c

?1
的最小值为

4。

ca

13.二次函数 y ? f (x) 的导函数 f ?(x) ? 2x ? m ,且 f (0) ? m 2 ?m ,则 f (x) ?0 在 R 上恒成立时

m 的取值范围是

. (??, 0) ( 4 , ??) 3

14.(理)已知函数 f (x) ? kx2 ? (k ? 3)x ? 1 的图象与 x 轴的交点在原点的右侧,则实数 k 的取值

范围是[0,1] .
(文)从正整数数列1,2,3,4,5,? 中删去所有的平方数,得到一个新数列,则这个新数列的第 1965
项是 2019 .

15、集合 A ? ??x | ?

x x

? ?

1 1

?

0?? ?

,

B

?

?x

||

x

?

b

|?

a?,若

“ a ? 1 ”是

“ A ? B ? ? ”的充分条件,求

b的

取值范围.

解: A ? ?x | ?1 ? x ? 1?,------------------------------------------------------------------------------4 分

B ? ?x | b ? a ? x ? a ? b?. a ? 1 ? A ? B ? ? ,∴ ? 2 ? b ? 2 .----------------------------9 分

∵ a ? 1 是 A ? B ? ? 的充分条件,只要 b 的取值范围是 (?2,2) 的子集即可.-------------14 分

16、二次函数 f (x) 满足 f (x ?1) ? f (x) ? 2x, 且 f (0) ? 1 . (Ⅰ)求 f (x) 的解析式; (Ⅱ)在区间??1,1? 上, y ? f (x) 的图象恒在 y ? 2x ? m 的图象上方,试确定实数 m 的范围.
解:(Ⅰ)设 f (x) ? ax2 ? bx ? c ,由 f (0) ? 1 得 c ? 1 ,故 f (x) ? ax2 ? bx ?1 .
∵ f (x ?1) ? f (x) ? 2x ,∴ a(x ?1)2 ? b(x ?1) ?1? (ax2 ? bx ?1) ? 2x .



2ax

?

a

?

b

?

2x

,所以

?2a?2 ??a?b?0

,

?a?1 ???b??1

,∴

f

(x)

?

x2

?

x ?1.

(Ⅱ)由题意得 x2 ? x ?1 ? 2x ? m 在[-1,1]上恒成立.即 x2 ? 3x ?1? m ? 0 在[-1,1]

上恒成立.



g(x)

?

x2

? 3x

?1?

m

,其图象的对称轴为直线

x

?

3 2

,所以

g(x)

在[-1,1]上递减.

故只需 g(1) ? 0 ,即12 ? 3?1?1? m ? 0 ,解得 m ? ?1 .

17、已知 f (x) ? ax3 ? 2ax2 ? b 在区间??2,1? 上最大值是 5,最小值是-11,求 f (x) 的解析式.

解 f (x) ? ax3 ? 2ax2 ? b,? f ' (x) ? 3ax2 ? 4ax ? ax(3x ? 4)



f

' (x) =0,得

x1

?

0,

x2

?

4 3

???2,1?

若 a >0,

………2

??2, 0?

0

?0,1?

f '(x)

+

0

-

f (x)



极大



因此 f(0)必为最大值,∴f(0)=5,得 b=5,

………………5

f (?2) ? ?16a ? 5, f (1) ? ?a ? 5,? f (1) ? f (?2)

? f (?2) ? ?16a ? 5 ? ?11,?a ? 1 ? f (x) ? x3 ? 2x2 ? 5;
若 a<0,同理可得 f(0)为最小值, ∴f(0)=-11,得 b=-11,
f (?2) ? ?16a ? 5, f (1) ? ?a ? 5,? f (?2) ? f (1)

……………8

? f (?2) ? f (x) max ? 5,?a ? ?1 ? f (x) ? ?x3 ? 2x2 ?11

……(14 分)

18、某房地产开发商投资 81 万元建一座写字楼,第一年装修费为 1 万元,以后每年增加 2 万元,把 写字楼出租,每年收入租金 30 万元.
(1)若扣除投资和装修费,则从第几年开始获取纯利润? (2)若干年后开发商为了投资其他项目,有两种处理方案:①纯利润总和最大时,以 10 万元出
售;②该楼年平均利润最大时以 46 万元出售该楼,问哪种方案更优? 解:(1)设第 n 年获取利润为 y 万元

n 年共收入租金 30n 万元,付出装修费构成一个以 1 为首项,2 为公差的等差数列,共
n ? n(n ?1) ? 2 ? n2 …………………………2 分 2

因此利润 y ? 30n ? (81 ? n2 ) ,令 y ? 0 ……………………3 分

解得: 3 ? n ? 27 ,…………………………………….4 分
所以从第 4 年开始获取纯利润 ………………………….5 分

(2)纯利润 y ? 30n ? (81 ? n2 ) ? ?(n ?15)2 ? 144

所以 15 后共获利润:144+ 10=154 (万元)………………………7 分

年平均利润W ? 30n ? (81 ? n2 ) ? 30 ? 81 ? n …………………..9 分

n

n

? 30 ? 2 81 ? 12 (当且仅当 81 ? n ,即 n=9 时取等号)……..10 分 n
所以 9 年后共获利润:12? 9 ? 46 =154(万元)………………….11 分
两种方案获利一样多,而方案②时间比较短,所以选择方案②……………12 分

19、已知函数

f

(x)

?

?x

?

log2

1? 1?

x x

,

(1)求 f ( 1 ) ? f (? 1 ) 的值;

2008

2008

(2)当 x ? ?? a, a?,其中a ? ?0,1?, a是常数,函数f (x) 是否存在最小值?若存在,求出 f (x) 的

最小值;若不存在,请说明理由。

1? x ? 0得(x ?1)(x ?1) ? 0解得 ?1 ? x ? 1 解:(文科)(1)由 1? x

函数 f (x) 的定义域为(-1,1)



f

(?x)

?

x

?

log2

1? 1?

x x

?

x

?

log2

1? 1?

x x

?

?

f

(x)

?函数f ( x为) 奇函数,即 f ?( x)? f ( x?) 0

? f ( 1 ) ? f (? 1 ) ? 0

2008

2008 …………………………………(6 分)

(2)任取 x1 、 x2 ? (?1,1)且设x1 ? x2.则

f

(x2 ) ?

f

( x1 )

?

( x1

?

x2 )

?

log2

1? 1?

x2 x2

? log2

1? 1?

x1 x1

?1 ? x1 ? x2 ? 1

? x1 ? x2 ? 0, 0 ? 1? x2 ? 1? x1,1? x2 ? 1? x1 ? 0

?1? 1?

x1 x1

?

1? x2 1? x2

即log2

1? 1?

x2 x2

? log2

1? 1?

x1 x1

?

0

? f (x2 ) ? f (x1) ? 0 即f (x2 ) ? f (x1)

?函数f (x)在(-1,1)上为减函数

又 x?(?a, a?且a?(0,1?

?函数f ( x在) ?( a?,上a 为) 减函数,故 fm i(xn ?)

f ?(a?)

?11a??2

a a

log
……(13

分)

20、已知函数 y ? f (x) ? ln x 。 x
(1)求函数 y ? f (x) 的图像在 x ? 1 处的切线方程;(2)求 y ? f (x) 的最大值; e
(3) 设实数 a ? 0 ,求函数 F(x) ? af (x) 在 ?a,2a?上的最小值

解(1)? f (x) 定义域为 ?0,???

1分

? f / (x) ? 1- lnx

3分

x2

? f (1) ? ?e

4分

e

又 ? k ? f / (1) ? 2e2

5分

e

?函数 y ? f (x) 的在 x ? 1 处的切线方程为: e

y ? e ? 2e2 (x ? 1) ,即 y ? 2e2 x ? 3e

6分

e

(2)令 f / (x) ? 0 得 x ? e

?当 x ? (0, e) 时, f / (x) ? 0 , f (x) 在 (0, e) 上为增函数

8分

当 x ? (e,??) 时, f / (x) ? 0 ,在 (e,??) 上为减函数

10 分

?

f m ax ( x)

?

f (e)

?

1 e

(3)? a ? 0 ,由(2)知:

F(x) 在 (0, e) 上单调递增,在 (e,??) 上单调递减。

? F(x) 在 ?a,2a?上的最小值 fmin (x) ? min{ F(a), F(2a)}

? F (a) ? F(2a) ? 1 ln a 22

?当 0 ? a ? 2 时, F(a) ? F(2a) ? 0, fmin (x) ? F(a) ? ln a

当2

?

a

时 F(a) ?

F (2a)

?

0,

f min (x)

?

F (2a)

?

1 ln 2

2a

12 分
13 分 14 分 15 分 16 分

一、填空题。

高三数学周周练(六)命题:郑介宏

1.函数 y ? log2 x ? 2 的定义域为_____________。

2.已知集合 A ? ?x | x ? a?,B ? ?x |1? x ? 2? ,且 A ? (CR B) ? R ,则实数 a 范围为:



3.设函数 f0 (x) ?| x |, f1 (x) ?| f0 (x) ?1 |, f2 (x) ?| f1(x) ? 2 | ,则函数 y ? f2 (x) 的图象与 x 轴所围

成的图形中封闭部分的面积为

.

4.二次函数 y ? x2 ? ax ?1 在区间[0,3] 上有最小值 ?2, 则实数 a 的值为

.

5.函数 y ? cos x 的值域是___________________. 2cos x ?1
6.设函数 f (x) ? n ?1, x ?[n, n ?1), n ? N ,则满足方程 f (x) ? log2 x 的根的个数是_________。

7.设

f (x)

?

?21?x ?

? a,

x

?0

,若 f (x) ? x 有且只有两个实数根,则实数 a 的取值范围是

.

? f (x ?1), x ? 0

8.给出四个命题:①函数 y ? a|x|与y ? loga | x | (a ? 0, a ? 1) 的图象关于直线 y ? x 对称;②函数

y ? a|x|与y ? ( 1 )|x| a

(a ? 0, a ? 1)

的图象关于

y

轴对称;③函数

y ? l o ag x | 与|

y ? log1 | x | (a ? 0, a ? 1) 的图象关于 x 轴对称,其中正确的命题是

.

a

9.已知 f (x) ? (x ? 1)? | x ? 1 |,若关于 x 的方程 f (x) ? x ? m 有三个不同的实数解,则实数 m 的

取值范围是

.

10.函数 f

(x)=

?lg | ??1(x

x ?

?2 2)

|

(

x

?

2)

若关于 x 的方程[f (x)] 2 +b·f (x)+C=0,恰有 3 个不

同的实数解 x1、x2、x3,则 f (x1+x2+x3)等于

。lg4

11.(理)直线 y ? a 与函数 f (x) ? x3 ? 3x 的图像有相异的三个公共点,则 a 的取值范围是



(文)一个等比数列的公比 q 满足 0 ? q ? 1,且它的第 4 项与第 8 项之和为 17 ,而第 5 项与第 7 8

项之积等于

1 4

.若它的前 n 项的和 Sn

?

31,则满足条件的自然数 n 的最小值为

.

12.已知二次函数

f

(x)

?

ax2

?

2x

?

c(x ?

R)

的值域为 [0,

??)

,则

a

?1

?

c

?1
的最小值为



ca

13.二次函数 y ? f (x) 的导函数 f ?(x) ? 2x ? m ,且 f (0) ? m 2 ?m ,则 f (x) ?0 在 R 上恒成立时

m 的取值范围是

.

14.(理)已知函数 f (x) ? kx2 ? (k ? 3)x ? 1 的图象与 x 轴的交点在原点的右侧,则实数 k 的取值

范围是



(文)从正整数数列1,2,3,4,5,?中删去所有的平方数,得到一个新数列,则这个新数列的第 1965

项是

.

二、解答题。

15、集合 A ? ??x | ?

x x

? ?

1 1

?

0?? ?

,

B

?

?x

||

x

?

b

|?

a?,若

“ a ? 1 ”是

“ A ? B ? ? ”的充分条件,求

b的

取值范围.

16、二次函数 f (x) 满足 f (x ?1) ? f (x) ? 2x, 且 f (0) ? 1 . (Ⅰ)求 f (x) 的解析式;
(Ⅱ)在区间??1,1? 上, y ? f (x) 的图象恒在 y ? 2x ? m 的图象上方,试确定实数 m 的范围.

17、已知 f (x) ? ax3 ? 2ax2 ? b 在区间??2,1? 上最大值是 5,最小值是-11,求 f (x) 的解析式.

19、已知函数

f

(x)

?

?x

?

log2

1? 1?

x x

,

(1)求 f ( 1 ) ? f (? 1 ) 的值;

2008

2008

(2)当 x ? ?? a, a?,其中a ? ?0,1?, a是常数,函数f (x) 是否存在最小值?若存在,求出 f (x) 的

最小值;若不存在,请说明理由。

18、某房地产开发商投资 81 万元建一座写字楼,第一年装修费为 1 万元,以后每年增加 2 万元,把 写字楼出租,每年收入租金 30 万元.
(1)若扣除投资和装修费,则从第几年开始获取纯利润? (2)若干年后开发商为了投资其他项目,有两种处理方案:①纯利润总和最大时,以 10 万元出
售;②该楼年平均利润最大时以 46 万元出售该楼,问哪种方案更优?

20、已知函数 y ? f (x) ? ln x 。 x
(1)求函数 y ? f (x) 的图像在 x ? 1 处的切线方程;(2)求 y ? f (x) 的最大值; e
(3) 设实数 a ? 0 ,求函数 F(x) ? af (x) 在 ?a,2a?上的最小值


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