问题 解答题

设函数f(x)=(x-a)2lnx,a∈R,e为自然对数的底数,e=2.7182…,如果对任意的x∈(0,3e],恒有f(x)≤4e2成立,求a的取值范围.

答案

f'(x)=(x-a)(2ln x+1-

a
x
).

①当0<x≤1时,对于任意的实数a,恒有f(x)≤0<4e2成立

②当1<x≤3e时,由题意,首先有f(3e)=(3e-a)2ln3e≤4e2,解得3e-

2e
ln3e
≤a≤3e+
2e
ln3e

由(I)知f′(x)=2(x-a)lnx+

(x-a)2
x
=(x-a)(2lnx+1-
a
x
),

令h(x)=2lnx+1-

a
x
,则h(1)=1-a<0,h(a)=2lna>0且h(3e)=2ln3e+1-
a
3e
≥2ln3e+1-
3e+
2e
ln3e
3e
=2(ln3e-
1
3
ln3e
)>0

又h(x)在(0,+∞)内单调递增,所以函数h(x)在(0,+∞)内有唯一零点,记此零点为x0

则1<x0<3e,1<x0<a,从而,当x∈(0,x0)时,f′(x)>0,当x∈(x0,a)时,f′(x)<0,当x∈(a,+∞)时,f′(x)>0,即f(x)在(0,x0)内是增函数,在(x0,a)内是减函数,在(a,+∞)内是增函数

所以要使得对任意的x∈(0,3e],恒有f(x)≤4e2成立只要有

f(x0)=(x0-a)2lnx04e2
f(3e)=(3e-a)2ln3e≤4e2

有h(x0)=2lnx0+1-

a
x0
=0得a=2x0lnx0+x0,将它代入f(x0)=(x0-a)2lnx04e2得4x02ln2x0≤4e2

又x0>1,注意到函数4x2ln2x在(1,+∞)上是增函数故1<x0≤e

再由a=2x0lnx0+x0,及函数2xlnx+x在(1,+∞)上是增函数,可得1<a≤3e

由f(3e)=(3e-a)2ln3e≤4e2解得3e-

2e
ln3e
≤a≤3e+
2e
ln3e

所以得3e-

2e
ln3e
≤a≤3e

综上,a的取值范围为3e-

2e
ln3e
≤a≤3e.

阅读理解

We find that bright children are rarely held back by mixed-ability teaching. On the contrary, both their knowledge and experience are enriched. We feel that there are many disadvantages in streaming(把…按能力分班)pupils. It does not take into account the fact that children develop at different rates. It can have a bad effect on both the bright and the not-so-bright child. After all, it can be quite discouraging to be at the bottom of the top grade!

  Besides, it is rather unreal to grade people just according to their intellectual (智力的) abilities. This is only one aspect of their total personality. We are concerned to develop the abilities of all our pupils to the full, not just their academic ability. We also value personal qualities and social skills, and we find that mixed-ability teaching contributes to all these aspects of learning.

  In our classrooms, we work in various ways. The pupils often work in groups: this gives them the opportunity to learn to co-operate, to share, and to develop leadership skills. They also learn how to cope with personal problems as well as how to think, to make decisions, to analyze and evaluate, and to communicate effectively. The pupils learn from each other as well as from the teacher.

  Sometimes the pupils work in pairs; sometimes they work on individual tasks and assignments, and they can do this at their own speed. They also have some library, and we teach them the skills they need in order to do this effectively. And expect our pupils to do their best, not their least, and we give them every encouragement to attain this goal.

1. Group work provides pupils with the opportunity_________.

  A. to develop academic abilities                    B. to learn to teach.

  C. to do some experiments                           D. to learn to be capable organizers.

2. By "held back" in the first paragraph the author means _________.

  A. drawn to their studies                                  B. prevented from advancing

  C. made to remain in the same classes            D. forced to study in the lower classes

3. In the passage the author's attitude towards " mixed-ability teaching" is ________.

  A.questioning               B. approving         C. objective    D. critical

4. The author's purpose of writing this passage is to _________.

  A. offer advice on the proper use of the library

  B. emphasize the importance of appropriate formal classroom teaching

  C. argue for teaching bright and not -so-bright pupils in the same class

  D. recommend pair work and group-work for classroom activities

5. The author argues that a teacher's chief concern should be the development of the student's _____.

  A. total personality                                   B. intellectual ability

  C. learning ability and communicative skills        D. personal qualities and social skills

填空题