问题 解答题
已知f(x)=
x2-6x-3
x+1
,g(x)=x3-3a2x-2a(a≥1),且它们定义域均为[0,1]
(1)求函数f(x)的最小值;
(2)判断函数g(x)的单调性并予以证明;
(3)若对任意t∈[0,1],总有g(x)≤f(t)在x∈[0,1]时恒成立,求a的取值范围.
答案

(1)由题意,f/(x)= 

x2+2x-3
(x+1)2

f/(x)=

x2+2x-3
(x+1)2
=0得x=-3或x=1

∵函数定义域为[0,1]

∴x=1时,函数f(x)的最小值-4;

(2)g′(x)=3x2-3a2=3(x+a)(x-a)

∵函数定义域为[0,1],a≥1

∴函数g(x)的单调减区间是[0,1],

(3)由(1)知,函数f(x)的最小值为-4,所以问题等价为 x3-3a2x-2a≤-4(a≥1),在x∈[0,1]时恒成立  

由(2)知,x=0时,函数g(x)取得最大值,所以-2a≤-4,故a≥2.

完形填空

Where Are We Going, Dad? Presents a new generation of men who, in a break from Chinese tradition, now take     active role in their children's lives. Five fathers and their children traipse(走过)   China, riding camels    the western deserts, fishing off the east coast, and selling vegetables for their bus fare home in Yunnan province. One dad doesn’t know    his daughter’s hair. Another one must survive (生存)with his son for three days in the desert, where, because   can cook, the two only eat instant noodles(泡面).

The children—and their bumbling(笨拙的)fathers—show remarkable candor(相对坦率). “I’m best at washing up(洗碗), I can’t do           ,” one dad says to another. “My wife is great—she’s been raising(养育) our son for six years. I’m       and it’s only been three days. I’m buying her some flowers when we       back,” says another.

But on the show, we see fathers who are         gentler on their kids and more involved in their upbringing.

These story lines are part of Where Are We Going Dad? Which        one of China’s most popular TV shows.

小题1:A. a                B. an               C. /

小题2:A. around           B. around the        C. all around

小题3:A. across            B. through          C. cross

小题4:A. how to do         B. what to do        C. how to do it

小题5:A. both             B. either            C. neither

小题6:A. something else     B. anything else      C. nothing else

小题7:A. exhausted         B. exhaust          C. exhausting

小题8:A. are going         B. will go           C. go

小题9:A. more           B. a lot of           C. a lot

小题10:A. has become       B. have become      C. has been become

单项选择题