问题 解答题

已知不等式ax2+bx+c>0的解集为(1,t),记函数f(x)=ax2+(a-b)x-c.

(1)求证:函数y=f(x)必有两个不同的零点.

(2)若函数y=f(x)的两个零点分别为m,n,求|m-n|的取值范围.

(3)是否存在这样实数的a、b、c及t,使得函数y=f(x)在[-2,1]上的值域为[-6,12].若存在,求出t的值及函数y=f(x)的解析式;若不存在,说明理由.

答案

(1)由题意知,∵a+b+c=0,且-

b
2a
>1,∴a<0且
c
a
>1
,∴ac>0.

对于函数f(x)=ax2+(a-b)x-c.有△=(a-b)2+4ac>0,∴f(x)必有2个不同零点.

(2)|m-n|2=(m+n)2-4mn=

(b-a)2+4ac
a2
=
(-2a-c)2+4ac
a2
=(
c
a
)2+8•
c
a
+4

由不等式ax2+bx+c>0的解集为(1,t)可知,ax2+bx+c=0的两个解分别为1和t(t>1),

由韦达定理有

c
a
=t,∴|m-n|2=t2+8t+4=(t+4)2-12,t∈(1,+∞),∴|m-n|2>52-12=13,∴|m-n| > 
13

即|m-n|的取值范围为(

13
,+∞).

(3)假设存在满足题意的实数a、b、c及t,∴f(x)=ax2+(a-b)x-c=a[x2+(1-

b
a
)x-
c
a
]=a[x2+(1+
a+c
a
)x-
c
a
]

=a[x2+(2+t)x-t](t≥1),∴f(x)的对称轴为x=-1-

t
2
<-
3
2
,∴f(x)在[-2,1]的最小值为f(1)=3a=-6,则a=-2.

要使函数y=f(x)在[-2,1]上的值域为[-6,12],只要f(x)max=12即可.

①若-1-

t
2
≤-2   ,  即t≥2时,f(x)max=f(-2)=123,则有6t=12,∴t=24.

此时,a=-2,b=6,c=-4,t=2,∴f(x)=-2x2-8x+4.

②若-1-

t
2
>-2   ,  ∴1<t<2,此时,f(x)max=f(-1-
t
2
)=
t2+8t+4
2
=12
,∴t=2(舍去),或t=-10(舍去 ).

综上所述:当a=-2,b=6,c=-4,t=2时,函数y=f(x)在[-2,1]上的值域为[-6,12],此时函数的表达式为f(x)=-2x2-8x+4.

阅读理解

阅读理解。

     If you do not use your arms or your legs for some time, they become weak; when you start

using them again, they slowly become strong again. Everybody knows this, and nobody would

think of questioning this fact.   1   When someone says that he has a good memory, he really

means that he keeps his memory in practice by exercising it regularly, either consciously or

unconsciously. When someone else says that his memory is poor, he really means that he does

not give it enough opportunity to become strong.  2   . One of them exercises his arms and legs

by playing tennis, while the other sits in a chair or a motor car all day.

     If a friend complains that his arms are weak, we know that it is his own fault. But if he tells us

that he has a poor memory, many of us think that his parents are to blame, or that he is just unlucky,

and few of us realize that it is just as much his own fault as if it was his arms or legs that were weak.

   3  . But all of us can, if we have ordinary bodies and brains, improve our strength and our memory

by the same means  practice.

     Have you ever noticed that people who cannot read or write usually have better memories than

those who can?  4   . Of course, because those who cannot read or write have to remember things.

They cannot write them down in a little notebook and they have to remember dates, time and prices,

names, songs and stories, so their memory is the whole time being exercised.

  5  .

A. What do you think of it?

B. Yet many people do not seem to know that the memory works in the same way.

C. Not all of us can become extremely strong or extremely clever.

D. So if you want a good memory, practice remembering.

E. Someone else says that he is poor in health.

F. Why is this?

G. The position is exactly the same as that of two people.

问答题 简答题