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问题 解答题

设a为正实数，函数f（x）=x3-ax2-a2x+1，x∈R．

（Ⅰ）求f（x）的极值；

（Ⅱ）设函数y=f（x）至多有两个零点，求实数a的取值范围．

答案

（Ⅰ）由f（x）=x3-ax2-a2x+1，得f'（x）=3x2-2ax-a2．（2分）

令f'（x）=3x2-2ax-a2=0，得x1=-

a
3
，x2=a(a＞0)，

x(-∞，-
a
3
)		-
a
3
(-
a
3
，a)		a（a，+∞）
f（x）+0-0+
 f（x）增极大减极小增
（5分）

∴f(x)极大=f(-

a
3
)=(-
a
3
)3-a(-
a
3
)2-a2×(-
a
3
)+1=
5
27
a3+1（6分）

f(x)极小=f(a)=a3-a3-a3+1=1-a3（7分）

（Ⅱ）由（Ⅰ）可得f（x）在(-∞，-

a
3
)上递增，在(-
a
3
，a)上递减，在（a，+∞）上递增，

f(x)极大=f(-

a
3
)=
5
27
a3+1＞0(a＞0)，f(x)极小=f(a)=a3-a3-a3+1=1-a3（9分）

当极小值f（a）=1-a3≥0，即0＜a≤1时，y=f（x）在x∈(-

a
3
，+∞)上有1个或0个零点，

此时f（-1）=a2-a=a（a-1）≤0，∴y=f（x）在x∈(-∞，-

a
3
)上有1个零点，

∴0＜a≤1时，y=f（x）有1个或2个零点；                         （11分）

当极小值f（a）=1-a3＜0，即a＞1时，y=f（x）在x∈(-

a
3
，+∞)上有2个零点，

此时f（-a）=1-a3＜0，y=f（x）在x∈(-∞，-

a
3
)上有1个零点，

∴当a＞1时，y=f（x）有3个零点；                                 （13分）

综上，若函数y=f（x）至多有两个零点，则a的取值范围是a∈（0，1]．（14分）

选择题
已知函数f（x）=
2x-x2
的定义域为A，则不等式组
x∈A
y-2≤0
x-y≤1
所表示的平面区域的面积为（　　）
A．7B．4C．3D．2
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单项选择题

Passage Four

In considering how the American family is changing, the starting point is the traditional family, a form which has developed over time on the basis of a number of assumptions. It is assumed that the family is heterosexual (异性的) institution, with prescriptions about how a man and a woman ought to be joined together and live together. The proper family form is assumed to be the nuclear family, that is, a family composed of a married man and woman and their children. It is assumed that the husband is the head of the family, with ultimate authority over wife and children, and that in their clearly separated roles the husband is the income-earner and the wife is the homemaker and provider of child care. It is assumed that the family lives by itself in its own house or residence.
Observers of family life have suggested that numerous changes are taking place in this traditional American family form. Included are the following general observations about trends.
More men and women seem to be living together before getting married.
Women and men seem to be marrying at a later age.
Married couples are having fewer children.
Unmarried women appear to be having more children.
Wives, even mothers with small children, are increasingly likely to be employed outside the home.
Marriages are more likely to end with divorce.
Single-parent families are more prevalent.
Remarriage is likely to follow divorce rather than widowhood.
Remarriage rates are declining, especially for women.
Some trends, viewed in a long-range historical context, appear not to be trends at all. The age at which men and women normally marry is the same now as it was 100 years ago, though it has fluctuated (变动，波动); a short-range increase in the 1940s and 1950s due to widespread postponement of marriage in war time made it appear that a change had occurred. Other trends appear to be following established patterns rather than representing a sharp break with tradition. Divorce rates have been increasing and families have been having fewer children for well over a century. Single-parent families and stepfamilies were very common in the past, although the reason for them was different. Many marriages formerly are dissolved and followed by remarriage because of the death of a parent or partner. Today single parenthood and remarriage are more likely to be a response to divorce.
Some social scientists see in these changes the breakdown of the family, to the detriment of the society. We take the position that family institutions and systems, like all human systems, are open, take many forms, and are constantly changing.

Which of the following is observed as a trend in the traditional family form

A．Family tends to get smaller.

B．People seem to get married younger.

C．More men tend to return home.

D．Remarriage rates tend to rise.
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