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www.vns0488.com:广西陆川县中学高二上学期数学同步作业:第6章 不等式 算术平均数与几何平均数(1)(大纲版)

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算术平均数与几何平均数(1)
一、选择题 1. a ? 0, b ? 0 是 a ? b ? 2 ab 的 A.充分不必要条件 B.必要不充分条件
1 2.设 a , b 为非零实数,则在○

C.充要条件 D.既不充分也不必要条件

a?b ab a 2 ? b2 a ? b 2 a 2 ? b2 2 ( 3 ? ? ab ;○ ) ? ;○ 中恒 2 a?b 2 2 2
C.2 D.3

成立的个数是 A.0

B.1

3.若 x, y ? R ,则下列不等式中最小值为 2 的是 A. a ? b
2 2

B. 2 ab

C. 2ab

D. a ? b

4.设 a, b ? (0,1) ,且 a ? b ,下列各式中最大的是 A. a ? 1 ? 2a
2

B. a ? 4 ? 4a
2

C. |

a b ? |? 2 b a

D.

2 a 2b 2 ? ab a 2 ? b2

二、填空题 5.设 a, b, c ? R ? , ab ? 2 ,且 c ? a ? b 恒成立,则 c 的最大值为
2 2

。 。

6.设 a ? 0, b ? 0, a, b 是常数,则当 x ? 0 时,函数 f ( x) ? 7.若 x ? R ,则
?
?

( x ? a)( x ? b) 的最小值是 x


x 的最 x ?1
2

值是

8.设 a, b ? R ,则 ab ,

a ? b a 2 ? b2 2 ,按从小到大顺序排列是 , , 1 1 2 2 ? a b



三、解答题 9.若 a ? 0, b ? 0 ,且

4 1 ? ? 1 ,求证: a ? 2b ? 6 ? 4 2 。 a b

10.若 a, b, c ? R ,求证:

?

1 1 1 1 1 1 ? ? ? ? ? 。 2a 2b 2c b ? c c ? a a ? b

11.若 a, b, c ? 0 ,求证: abc ? (a ? b ? c)(b ? c ? a)(c ? a ? b) 。

参考答案 一、选择题 1.A 2.C 二、填空题 5. 4 6. ( a ? b )2

3.D

4.D

7.大;

1 2

8.

2 1 1 ? a b

? ab ?

a?b a 2 ? b2 ? 2 2

9. 略 10.略(提示:

1 1 1 1 1 ( ? )? ? ) 2 2a 2b 2 ab a ? b

11.不妨设 a ? b ? c ? 0 ,然后对 b ? c ? a ? 0 与 b ? c ? a ? 0 进行分类证明即可。


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