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2019-2020学年高中数学 第1部分 1.3.1 第二课时 函数的最大(小)值应用创新演练 新人教A版必修1.doc.doc

2019-2020 学年高中数学 第 1 部分 1.3.1 第二课时 函数的最大(小) 值应用创新演练 新人教 A 版必修 1
1.若函数 y=ax+1 在[1,2]上的最大值与最小值的差为 2,则实数 a 的值是( A.2 C.2 或-2 B.-2 D.0 )

解析:a>0 时,由题意得 2a+1-(a+1)=2,即 a=2;

a<0 时,a+1-(2a+1)=2,∴a=-2.
综上,a=±2. 答案:C
? ?2x+6,x∈[1,2], 2.函数 f(x)=? ?x+7,x∈[-1, , ?

则 f(x)的最大值、最小值分别为(

)

A.10,6 C.8,6

B.10,8 D.以上都不对

解析:f(x)在[-1,2]上单调递增, ∴最大值为 f(2)=10,最小值为 f(-1)=6. 答案:A 3.已知函数 f(x)=-x +4x+a,x∈[0,1],若 f(x)有最小值-2,则 f(x)的最大值为 ( ) A.-1 C.1
2 2

B.0 D.2
2

解析:∵f(x)=-(x -4x+4)+a+4=-(x-2) +4+a, ∴函数 f(x)图象的对称轴为 x=2. ∴f(x)在[0,1]上单调递增. 又∵f(x)min=-2,∴f(0)=-2,即 a=-2. ∴f(x)max=f(1)=-1+4-2=1. 答案:C 4.某公司在甲乙两地同时销售一种品牌车,销售 x 辆该品牌车的利润(单位:万元)分 别为 L1=-x +21x 和 L2=2x.若该公司在两地共销售 15 辆,则能获得的最大利润为( A.90 万元 C.120 万元 B.60 万元 D.120.25 万元
2 2

)

解析:设公司在甲地销售 x 辆,则在乙地销售(15-x)辆,公司获利为 L=-x +21x +2(15-x)

19 2 19 2 =-x +19x+30=-(x- ) +30+ , 2 4 ∴当 x=9 或 10 时,L 最大为 120 万元. 答案:C 5. 函数 f(x)在[-2,2]上的图象如图所示, 则此函数的最小值、 最大值分别是________、________. 解析:由图象可知,最高点为(1,2),最低点为(-2,-1). 因此,最小值为-1,最大值为 2. 答案:-1 2 6.函数 f(x)=

2

x 在区间[2,4]上的最大值为______,最小值为________. x+2 x+2-2 2 = =1- , x+2 x+2 x+2 x

解析:∵f(x)=

∴函数 f(x)在[2,4]上是增函数. 2 1 ∴f(x)min=f(2)= = , 2+2 2

f(x)max=f(4)=
2 1 答案: 3 2

4 2 = . 4+2 3

4 7.已知函数 f(x)=x+ ,x∈[1,3].

x

(1)判断 f(x)在[1,2]和[2,3]上的单调性; (2)根据 f(x)的单调性写出 f(x)的最值. 解:(1)设 x1,x2 是区间[1,3]上的任意两个实数,且 x1<x2, 4 4 4 则 f(x1)-f(x2)=x1-x2+ - =(x1-x2)(1- ).

x1 x2

x1x2

∵x1<x2,∴x1-x2<0. 当 1≤x1<x2≤2 时,1<x1x2<4,∴ ∴1- 4 <0.∴f(x1)>f(x2). 4

x1x2

>1.

x1x2

∴f(x)在[1,2]上是减函数. 当 2≤x1<x2≤3 时,4<x1x2<9, ∴0< 4

x1x2

<1.∴1-

4

x1x2

>0.

∴f(x1)<f(x2).

∴f(x)在[2,3]上是增函数. 4 (2)由(1)知 f(x)的最小值为 f(2)=2+ =4. 2 4 13 又∵f(1)=5,f(3)=3+ = <f(1), 3 3 ∴f(x)的最大值为 5. 8.已知函数 f(x)=x +2ax+2,x∈[-5,5]. (1)当 a=-1 时,求函数 f(x)的最大值和最小值; (2)求实数 a 的取值范围,使 y=f(x)在区间[-5,5]上是单调函数. 解:(1)当 a=-1 时,
2

f(x)=x2-2x+2=(x-1)2+1,x∈[-5,5].
当 x=1 时,有 f(x)min=1; 当 x=-5 时,有 f(x)max=37. (2)∵函数 f(x)=(x+a) +2-a 图象的对称轴为
2 2

x=-a,f(x)在区间[-5,5]上是单调函数,
∴-a≤-5 或-a≥5,即 a≥5 或 a≤-5. 所以 a 的取值范围为(-∞,-5]∪[5,+∞).


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