问题 问答题

确定方程1+lnx=kx的根的个数,其中常数k可取任何实数。

答案

参考答案:引入函数F(x)=1+lnx-kx,则对于任何实数k,函数F(x)都在区间(0,+∞)内可导,且[*]
由此可见,当k≤0时对任何x>0都有F’(x)>0,这表明当k≤0时函数F(x)在区间(0,+∞)内单调增加,又因[*],故当k≤0时函数F(x)在区间(0,+∞)内有唯一零点,即当k≤0时方程1+lnx=kx有唯一根,特别,当k=0时方程1+lnx=kx变成方程1+lnx=0,易见它有唯一根。[*]
当k>0时,由于
[*]
从而函数F(x)在区间[*]内单调增加,在区间[*]内单调减少,在点[*]处取得最大值[*]
当k>1时,由于函数F(x)的最大值-lnk<0,故这时方程1+lnx=kx无根;当k=1时,由于函数F(x)的最大值-lnk=0,故这时方程1+lnx=kx有唯一根x=1;而当0<k<1时,由于函数F(x)的最大值-lnk>0,故这时方程1+lnx=kx有两个根x1与x2,且[*]与[*]

单项选择题
阅读理解

阅读理解。

     The common cold is the world’s most widespread illness, which is plagues(疫病) that flesh receives.

     The most widespread mistake of all is that colds are caused by cold. They are not. They are caused by viruses passing on from person to person. You catch a cold by coming into contact, directly or indirectly, with someone who already has one. If cold causes colds, it would be reasonable to expect the Eskimos to suffer from them forever. But they do not. And in isolated Arctic regions explorers have reported being

free from colds until coming into contact again with infected people from the outside world by way of

packages and mail dropped from airplanes.

     During the First World War soldiers who spent long periods in the trenches(战壕), cold and wet,

showed no increased tendency to catch colds.

     In the Second World War prisoners at the notorious Auschwitz concentration camp(奥斯维辛集中营), naked and starving, were astonished to find that they seldom had colds.

     At the Common Cold Research Unit in England, volunteers took part in experiments in which they put

themselves into the discomforts of being cold and wet for long stretches of time. After taking hot baths, they put on bathing suits, allowed themselves to be with cold water, and then stood about dripping wet in

drafty room. Some wore wet socks all day while others exercised in the rain until close to exhaustion. Not one of the volunteers came down with a cold unless a cold virus was actually dropped in his nose.

     If, then, cold and wet have nothing to do with catching colds, why are they more frequent in the winter? Despite the most pains-taking research, no one has yet found the answer. One explanation offered by

scientists is that people tend to stay together indoors more in cold weather than at other times, and this

makes it easier for cold viruses to be passed on.

     No one has yet found a cure for the cold. There are drugs and pain suppressors(止痛片) such as

aspirin, but all they do is to relieve the symptoms.

1. The writer offered _______ examples to support his argument.

A. 4

B. 5

C. 6

D. 3

2. Which of the following does not agree with the chosen passage?

A. The Eskimos do not suffer from colds all the time.

B. Colds are not caused by cold.

C. People suffer from colds just because they like to stay indoors.

D. A person may catch a cold by touching someone who already had one.

3. Arctic explorers may catch colds when ________.

A. they are working in the isolated Arctic regions

B. they are writing reports in terribly cold weather

C. they are free from work in the isolated Arctic regions

D. they are coming into touch again with the outside world

4. The passage mainly discusses ________.

A. the experiments on the common cold

B. the false opinion about the common cold

C. the reason and the way people catch colds

D. the continued spread of common colds