问题 解答题
已知f(x)是二次函数,不等式f(x)<0的解集是(0,5),且f(x)在点(1,f(1))处的切线与直线6x+y+1=0平行.
(1)求f(x)的解析式;
(2)是否存在t∈N*,使得方程f(x)+
37
x
=0
在区间(t,t+1)内有两个不等的实数根?若存在,求出t的值;若不存在,说明理由.
答案

(1)∵f(x)是二次函数,且f(x)<0的解集是(0,5),

∴可设f(x)=ax(x-5)=ax2-5ax,(a>0).

∴f(x)在点(1,f(1))处的切线斜率是:f′(1)=-3a=-6.

∴a=2,∴f(x)=2x(x-5)=2x2-10x(x∈R).

(2)方程f(x)+

37
x
=0等价于方程 2x3-10x2+37=0.

设h(x)=2x3-10x2+37,则h'(x)=6x2-20x=2x(3x-10).

在区间x∈(0,

10
3
)时,h'(x)<0,h(x)是减函数;

在区间(-∞,0),或(

10
3
,+∞)上,h'(x)>0,h(x)是增函数,故h(0)是极大值,h(
10
3
)是极小值.

∵h(3)=1>0,h(

10
3
)=-
1
27
<0,h(4)=5>0,

∴方程h(x)=0在区间(3,

10
3
),(
10
3
,4)内分别有惟一实数根,故函数h(x)在(3,4)内有2个零点.

而在区间(0,3),(4,+∞)内没有零点,在(-∞,0)上有唯一的零点.

画出函数h(x)的单调性和零点情况的简图,如图所示.

所以存在惟一的正整数t=3,使得方程f(x)+

37
x
=0在区间(t,t+1)内有且只有两个不同的实数根.

阅读理解

Play is the basic business of childhood, and in recent years more and more research has shown the great importance of play in the development of a human being. From earliest  infancy (婴儿), every child needs opportunity and the right material for play, and the main tools of play are toys. The main function of toys is to suggest, encourage and assist play. To succeed in this, they must be good toys, which children will play with often and will come back to again and again. Therefore, it is important to choose suitable toys for different stages of a child’s development.

In recent years research on infant development has shown that the standard a child is likely to reach, within the range of his inherited(遗传的) abilities, is largely determined in the first three years of his life. So a baby's ability to benefit from the right play materials should not be underestimated. A baby who is encouraged, talked to and shown things and played with, has the best chance of growing up successfully.

The next stage, from three to five years old, curiosity knows no bounds. Every type of suitable toy should be made available to the child, for trying out, experimenting and learning, for discovering his own particular ability. Bricks and jigsaws and construction toys; painting, scribbling(乱涂) and making things; sand and water play; toys for imaginative and pretending play; the first social games for learning to play and get on with others.

By the third stage of play development—from five to seven or eight years old— the child is at school. But for a few more years play is still the best way of learning, at home or at school. It is easier to see which type of toys the child most enjoys.

Until the age of seven or eight, play and work mean much the same to a child. But once reading has been mastered, then books and school become the main source of learning. Toys are still interesting and valuable, which lead up to new hobbies, but their significance has changed —to a child of nine or ten years old, toys and games mean, as to adults, relaxation and fun.

小题1:The writer wants us to understand that a child_______.

A.cannot grow up without toys

B.matures(使成熟) through play

C.uses toys as friends

D.has to be taught how to play小题2:According to the passage, the abilities a child has inherited from his parents ________.

A.determine his character

B.will not change after the age of three

C.partly determine the standard he is likely to reach

D.to a large extent determine the choice of toys小题3:The passage tells us that children are the most curious when they are about______.

A.two years old

B.one year old

C.six years old

D.four years old小题4:The article is about_______.

A.the importance of play

B.the importance of books

C.the relationship between play and work

D.children’s speech development

判断题