问题 解答题

在函数y=lgx(x>1)的图象有A、B、C三点,横坐标分别为:m,m+2,m+4,

(1)若△ABC的面积为S,求S=f(m)的解析式;

(2)求S=f(m)的最大值;

(3)若a、b、c和lga、lgb、lgc分别是两个三角形的三条边,且a、b、c互不相等,那么这两个三角形能否相似?说明理由.

答案

(1)过分别过A,B,C作AA1⊥x轴,BB1⊥x轴,CC1⊥x轴分别交x轴于A1,B1,C1

则S=S梯形AA1B1B+S梯形BB1C1D-S梯形AA1C1D

=

1
2
[lgm+lg(m+2)]×2+
1
2
[lg(m+2)+lg(m+4)]×2-
1
2
[lgm+lg(m+4)]×4=2lg(m+2)-lg(m+4)-lgm(m≥1)

∴S△ABC=lg

(m+2)2
m(m+4)
=lg(1+
4
m2+4m
),(m≥1)

(2)S△ABC=lg(1+

4
(m+2)2-4
)这个函数可以看做复合函数

因为S=lgx是增函数,所以要是S最大,只要函数x=1+

4
(m+2)2-4
最大即可

在[1,+∞)⇒Smgx=f(1)=lg

9
5
=2lg3-lg5;

(3)不相似.依题意,设a<b<c,则有lga<lgb<lgc,若相似,则有

a
lga
=
b
lgb
=
c
lgc
=k,k为常数

⇒a、b、c为方程x=klgx的三个根.

而曲线y=x与y=klgx至多有两个交点,所以产生矛盾,因而这两个三角形不相似.

单项选择题
阅读理解

Researchers have found more evidence that suggests a relationship between races (种族) and rates (率) of lung cancer among smokers. A new study shows that black people and Native Hawaiians are more likely to develop lung cancer from smoking. It compared their risk to whites, Japanese-Americans and Latinos.

Researchers at the University of Southern California and the University of Hawaii did the new study. The New England Journal of Medicine published the findings. The eight-year research studied more than 180,000 people. They included present and former smokers and people who never smoked. Almost 2.000 people in the study developed lung cancer.

Researchers say genetics (遗传学) might help explain the racial and ethnic(种族的) differences. There could be differences in how people's bodies react to smoke. But environmental influences, including the way people smoke, could also make a difference.

African-Americans and Latinos in the study are reported smoking the fewest cigarettes per day. Whites are the heaviest smokers. But the scientists point out that blacks have been reported to breathe cigarette smoke more deeply than white smokers. This could fill their lungs with more of the chemicals in tobacco that cause cancer.

Scientists know that some diseases effect different groups differently. And some drug companies have begun to develop racially targeted (针对) medicines. Last June, the United States Food and Drug Administration approved a drug designed to treat heart failure in black patients. The name is BiDil. The agency called it "a step toward the promise of personalized medicine."

小题1:Researchers agree that it is     that may probably determine black people’s risk of lung cancer.

A.the larger amount of smoking than white people

B.the living style or habit of the blacks

C.the depth of cigarette smoke into their lungs

D.the physical strength to react to cigarette smoke小题2:People in the new study are made up of         .

A.heavy smokers in America

B.smokers and non-smokers

C.the Asians and Hawaiians

D.the black and white people小题3:The production of BiDil referred to in the last paragraph is to     .

A.explain different races react to some diseases differently

B.tell the readers that racial differences exist in smokers

C.show a big step people have taken in the medicine area

D.support the idea that it is easy for blacks to have cancers小题4:Which of the following statements is TRUE according to the author?

A.The way of smoking may increase the risk of lung cancer.

B.Race has nothing to do with the risk of having a lung cancer.

C.The research was started by the New England Journal of Medicine.

D.he risk of lung cancer lies I how much a person smokes.