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weu107:2019学年人教A版高中数学必修五导学案:3.4.1基本不等式的应用(3)

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课题: 3.4.1 基本不等式(1)
班级: 【学习目标】 姓名: 学号: 第 学习小组

理解算术平均数与几何平均数的定义及它们的关系.探究并了解基本不等式的证明过 程,会用各种方法证明基本不等式.理解基本不等式的意义,并掌握基本不等式中取等 号的条件是:当且仅当这两个数相等. 【课前预习】 1.当 a, b 满足条件__________时,基本不等式

a?b ? ab 成立, 2

该不等式取符号的条件是____________________________________. 2.算术平均数的定义: 3.几何平均数的定义: 4.算术平均数与几何平均数的关系 (1)基本公式: ab ?

a?b 及语言叙述 2

(2)基本不等式的证明方法 (3)基本不等式成立的条件 (4)基本不等式的变形

【课堂研讨】
例 1.设 a, b 为正数,证明下列不等式: (1)

b a ? ? 2; a b

(2) a ?

1 ? 2. a

变化:若 a, b 都为负数,则分别比较

b a 1 ? 与 2 ; a ? 与 ? 2 的大小. a a b

2 2 例 2 若 a,b ? R,a ? b ,求证: a ? b ? 2a ? 2b ? 2 .

例 3.若 a, b 都是正整数,求证:

2ab a ? b ? . a?b 2

? ? ?

【学后反思】

课题:3.4.1 基本不等式的证明(1)检测案
班级: 【课堂检测】
1. 证明: (1) a ? b ? 2ab ;
2 2

姓名:
2

学号:
(2) x ? 1 ? 2 x ; (



学习小组
3 )

x?

1 ? ?2 ( x ? 0) . x

2.设 x,y ? R ,求证: x 2 ? 4 y 2 ? 2 ? 2 x ? 4 y .

2. 求证: (

a ? b 2 a2 ? b2 ) ? . 2 2

【课后巩固】
1.若 a ? b ? 1 , P ? A. R ? P ? Q D. P ? R ? Q 2.若 b ? a ? 0 ,则下列不等式一定成立的是( )

lg a ? lg b , Q ?

1 a?b (lg a ? lg b) , R ? lg ,则( 2 2 B. P ? Q ? R C. Q ? P ? R



a?b ? ab ? b A. a ? 2 a?b ? ab ? a C. b ? 2
2

a?b ?a 2 a?b ? ab D. b ? a ? 2
B. b ?

ab ?

3. (1) P ? (4 ? a )( 4 ?

1 ),Q ? 24 ,则 P 与 Q 的大小关系为_________. a2

(2)已知 a ? 1 ,则 P ?

1 a ?1 log 2 a 与 Q ? log 2 的大小关系为_________. 2 2

4.设 a , b ? (0, ???? ?) ,求证:

2ab ? ab . a?b

5.设 x,y ? R ,求证: x ? y ? 5 ? 2(2x ? y) .
2 2


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