问题 解答题

已知函数f(x)=x(x2-a),(a∈R)

(1)求f(x)的单调区间;

(2)若过点P(1,-2)可以向y=f(x)作两条切线,求a的取值范围.

答案

(1)由f(x)=x3-ax求导数得到f'(x)=3x2-a.

(i)当a≤0时,f'(x)≥0,则f(x)在R上单增.

(ii)当a>0时,f′(x)=3(x-

a
3
)(x+
a
3
).

f(x)在[

a
3
,+∞)和(-∞,-
a
3
]上单调递增;

[-

a
3
,+
a
3
]上单调递减.…(5分)

(2)设过P(1,-2)向y=f(x)作切线于切点(x0,y0),

则y-y0=(3x02-a)(x-x0),即y=(3x02-a)x-2x03

则y=(3x02-a)x-2x03过P(1,-2),

∴-2=3x02-a-2x03,即2x03-3x02+a-2=0.

由题意知关于x0的方程

2x03-3x02+a-2=0有两个不等的实根.

令g(x)=2x3-3x2+a-2,

则g'(x)=6x2-6x=6x(x-1).

于是g(x)极小=g(1)=a-3,

g(x)极大=g(0)=a-2.

方程g(x)=0有三个实根,其中两个根是等根.

g(1)=a-3=0
g(0)=a-2>0
g(1)<0
g(0)=0.

∴a=3或a=2

∴所求a的取值范围为[2,3].…(13分)

阅读理解

Welcome to the Ambassador Hotel. To make your stay as enjoyable as possible, we hope you can make full use of our facilities(设备).

Dining Room

Breakfast is served in the dining room from 8 a.m. to 9:30 a.m. The room staff will bring a breakfast tray to your room at any time after 7 a.m. Please fill out a card and hang it outside your door when you go to bed.

Lunch: 12:00 a.m. to 2:30 p.m.

Dinner: 7:30 p.m. to 9:00 p.m.

Room Service

This operates 24 hours a day; phone the Reception Desk, and your message will be passed on to the room staff.

Telephones

To make a phone call, dial 0 for Reception and ask to be connected. We apologize for delays in putting calls through when the staffs are very busy. There are also public telephone boxes near the Reception Desk. Early calls should be booked with Reception.

Laundry

We have a laundry in the hotel, and will wash, iron and return your clothes within 24 hours. Ask the room staff to collect them.

Bar

The hotel bar is open from 12 a.m. to 2 p.m. and 7 p.m. to 1 a.m.

小题1:You would see this notice ___________.

A.in a hotel bar

B.in a hotel dining room

C.in a bedroom of a large hotel

D.in the entrance of a small family hotel小题2:What time can you have breakfast in the dinning room?

A.6:50 a.m.

B.7:00 a.m.

C.7:30 a.m.

D.8:30 a.m.小题3:What should you do when you come back to the hotel to make an important call and see a lot of people around the Reception Desk?

A.Go to your room and phone from there.

B.Ask at the Reception Desk.

C.Go to the public telephone boxes.

D.Go out again and look for a private phone.小题4:What should you do if you want to have your trousers washed?

A.Ask the room staff to collect them to the laundry.

B.Go to the hotel bar.

C.Hang a message outside your door.

D.Ask the room staff to clean them for you.

单项选择题