高三函数练习题难题

1、设函数  

（1）求的单调区间、最大值；

（2）讨论关于的方程的根的个数．

2、已知函数f(x)的定义域是（0，+∞），当x>1时，f(x)>0，且f(x·y)＝f(x)+f(y)，

(1)求f(1)的值； 

(2)证明f(x)在定义域上是增函数；

(3)如果f( 3 )=1，求满足不等式f(x)－f()≥2的x的取值范围．

3、已知函数．

（I）当时，求函数的定义域；

（II）若关于的不等式的解集是，求的取值范围．

4、已知命题p：指数函数f(x)＝(2a－6)x在R上单调递减，命题q：关于x的方程x2－3ax＋2a2＋1＝0的两个实根均大于3.若p或q为真，p且q为假，求实数a的取值范围．

5、已知函数.

（Ⅰ）求函数的单调区间；

（Ⅱ）若函数上是减函数，求实数a的最小值；

（Ⅲ）若，使（）成立，求实数a的取值范围. 

6、

7、

(1)求常数c的值；

(2)解不等式f(x)>＋1.

二、填空题

8、下列四个命题，是真命题的有　　　　(把你认为是真命题的序号都填上).

①若在区间(1,2)上有一个零点；，则p∧q为假命题；

②当时，的大小关系是；

③若，则在处取得极值；

④若不等的解集为，函数的定义域为，则“”是“”的充分不必要条件.

9、已知函数f(x),g(x)分别是定义在R上的奇函数和偶函数，当x<0时，

  且g(3)=0,则不等式f(x)g(x)<0的解集是_               _.

10、已知为常数,若函数有两个极值点，则的取值范围是               

三、选择题

11、定义在R上的偶函数在上单调递减，且，则满足 的的 集合为      (　　 )

A. (－∞，)∪(2，＋∞)    B. (，1)∪(1,2)  C. (，1)∪(2，＋∞)     D. (0，)∪(2，＋∞)

12、已知是上的奇函数，且当时，，那么 的值为（   ）

A．0                 B．         C．            D．

13、已知是R上的偶函数，当时，是函数的零点，则的大小关系是(   )     

    A．          B．

    C．          D．

14、已知函数 若，使得成立，则实数的取值范围是（   ）         

A.       B.    C.       D.或

15、已知函数f(x)＝sinx＋ex＋x2 011，令f1(x)＝f′(x)，f2(x)＝f′1(x)，f3(x)＝f′2(x)，…，fn＋1(x)＝f′n(x)，则f2 012(x)＝                                                          (　　)

A．sinx＋ex                         B. cosx＋ex

C．－sinx＋ex                       D．－cosx＋ex 

四、综合题

（每空？ 分，共？ 分）

16、设函数f(x)＝ex－ax－2.

(1)求f(x)的单调区间；

(2)若a＝1，k为整数，且当x>0时，(x－k)f′(x)＋x＋1>0，求k的最大值．

参

一、简答题

1、解：（1）　　………………1分

　　　　由得

　　　　当时，，单调递增；

　　　　当时，，单调递减；

　　　　∴函数的单调递增区间是；单调递减区间是　…………3分

　　　　∴的最大值为　　…………4分

　　　　（2）令＝　…………5分

　　　　①当时，

　　　　∴

∵　　∴　　　　

∴在上单调递增　　　　　　　　　　　　………………7分

　　　　②当时，，

　　　　∵

　　　　∴　　　　　∴在（0,1）上单调递减　　

　　　综合①②可知，当时，　　　　　…………9分

　　　当即时，没有零点，故关于方程的根的个数为0

　　　当即时，只有一个零点，故关于方程的根的个数为1

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　……………………11分

　　　当即时，当时

　　　由（1）知

　　　要使，只需即

　　　　当时，　由(1)知

　　　　要使，只需即

　　　　所以时，有两个零点　　　………………13分

　　　　综上所述

　　　　当时，关于的方程根的个数为0

　　　　当时，关于的方程根的个数为1

　　　　当时，关于的方程根的个数为2　　…………14分

2、解：（1）令x=y=1，得f(1)=2f(1)，故f(1)=0. ………………………………2分

(2)令y= ,得f(1)=f(x)+f()=0,故f()=－f(x). ………………………………4分

任取x1,x2∈（0，+∞）,且x1由于>1，故f()>0，从而f(x1)>f(x2).  ∴f(x)在（0，+∞）上是增函数. …………8分

(3)由于f(3)=1，在f(x·y)＝f(x)+f(y)中，令x=y=3，得f(9)=f(3)+f(3)=2。……10分

又－f()=f(x－2)，故所给不等式可化为f(x)+f(x－2) ≥f(9)，即 f [x(x－2)] ≥f(9)，

解得.∴x的取值范围是.…………………14分

3、解：（I）由题设知：，        …………1分

不等式的解集是以下三个不等式组解集的并集：              

，或，或，…………4分

解得函数的定义域为；               …………6分

（II）不等式即，                 …………8分

∵时，恒有，        …………10分

∵不等式解集是，

∴，求得的取值范围是．………12分

4、解：若p真，则f(x)＝(2a－6)x在R上单调递减，

∴0<2a－6<1，∴3若q真，令f(x)＝x2－3ax＋2a2＋1，则应满足

，∴，故a>，

又由题意应有p真q假或p假q真．

①     若p真q假，则，a无解．

②若p假q真，则，∴故a的取值范围是{a|5、解：由已知函数的定义域均为,且. …1分

（Ⅰ）函数,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        

当且时,;当时,.

所以函数的单调减区间是,增区间是.  ………………3分

（Ⅱ）因f(x)在上为减函数，故在上恒成立． 

所以当时，．

又，

故当，即时，．

所以于是，故a的最小值为．       …………………………6分

（Ⅲ）命题“若使成立”等价于

“当时，有”．         

由（Ⅱ），当时，，．  

问题等价于：“当时，有”．      ………………………8分

当时，由（Ⅱ），在上为减函数，

则=，故．       

当0<时，由于在上为增函数，

故的值域为，即．

由的单调性和值域知，

唯一，使，且满足：

当时，，为减函数；当时，，为增函数；

所以，=，．

所以，，与矛盾，不合题意．

综上，得．                         …………………………14分

6、解: (Ⅰ)当时,.

因为当时,,,

且,

所以当时,,且

由于,所以,又,

故所求切线方程为,

即

   (Ⅱ) 因为,所以,则

①     当时,因为,,

所以由,解得,

从而当时, 

②     当时,因为,,

所以由,解得,

从而当时, 

③     当时,因为,

    从而 一定不成立

综上得,当且仅当时,,

故

从而当时,取得最大值为

(Ⅲ)“当时,”等价于“对恒成立”,即“(*)对恒成立”

①     当时,,则当时,,则(*)可化为

,即,而当时,,

所以,从而适合题意

②     当时,.

⑴     当时,(*)可化为,即,而,

所以,此时要求

⑵     当时,(*)可化为,

所以,此时只要求

(3)当时,(*)可化为,即,而,

所以,此时要求，由⑴⑵⑶,得符合题意要求.

 综合①②知,满足题意的存在,且的取值范围是

7、解析　(1)∵0∴c＝.

(2)由(1)得f(x)＝

由f(x)>＋1，得当0当≤x<1时，解得≤x<.

∴f(x)>＋1的解集为. 

二、填空题

8、 ①②④

9、； 

10、   0三、选择题

11、D 

12、D 

13、C 

14、A  

15、A

解析　∵f1(x)＝f′(x)＝cosx＋ex＋2 011x2 010，f2(x)＝f′1(x)＝－sinx＋ex＋2 011×2 010x2 009，f3(x)＝f′2(x)＝－cosx＋ex＋2 011×2 010×2 009x2 008，f4(x)＝f′3(x)＝sinx＋ex＋2 011×2 010×2 009×2 008x2 007，…，∴f2 012(x)＝sinx＋ex.

四、综合题

16、

解析　(1)f(x)的定义域为(－∞，＋∞)，f′(x)＝ex－a.

若a≤0，则f′(x)>0，所以f(x)在(－∞，＋∞)上单调递增．

若a>0，则当x∈(－∞，lna)时，f′(x)<0；当x∈(lna，＋∞)时，f′(x)>0，所以，f(x)在(－∞，lna)上单调递减，在(lna，＋∞)上单调递增．

(2)由于a＝1，所以(x－k)f′(x)＋x＋1＝(x－k)(ex－1)＋x＋1.

故当x>0时，(x－k)f′(x)＋x＋1>0等价于

k<＋x(x>0)． ①

令g(x)＝＋x，则g′(x)＝＋1＝.

由(1)知，函数h(x)＝ex－x－2在(0，＋∞)上单调递增．而h(1)<0，h(2)>0，所以h(x)在(0，＋∞)上存在唯一的零点．故g′(x)在(0，＋∞)上存在唯一的零点．设此零点为α，则α∈(1,2)．

当x∈(0，α)时，g′(x)<0；当x∈(α，＋∞)时，g′(x)>0.所以g(x)在(0，＋∞)上的最小值为g(α)．又由g′(α)＝0，可得eα＝α＋2，所以g(α)＝α＋1∈(2,3)．

由于①式等价于k