问题 解答题

如果正实数a,b满足ab=ba.且a<1,证明a=b.

答案

证一:由ab=ba,得blna=alnb,从而

lna
a
=
lnb
b

考虑函数y=

lnx
x
(0<x<+∞),它的导数是y′=
1-lnx
x2
.

因为在(0,1)内f'(x)>0,所以f(x)在(0,1)内是增函数

由于0<a<1,b>0,所以ab<1,从而ba=ab<1.由ba<1及a>0,

可推出b<1.

由0<a<1,0<b<1,假如a≠b,

则根据f(x)在(0,1)内是增函数,

得f(a)≠f(b),即

lna
a
lnb
b

从而ab≠ba这与ab=ba矛盾

所以a=b

证二:因为0<a<1,ab=ba

所以blogaa=alogab,即

b
a
=logab

假如a<b,则

b
a
>1,但因a<1,

根据对数函数的性质,

logab<logaa=1,从而

b
a
>logab,这与
b
a
=logab矛盾

所以a不能小于b

假如a>b,则

b
a
<1,而logab>1,这也与
b
a
=logab
矛盾

所以a不能大于b,因此a=b

证三:假如a<b,则可设b=a+ε,其中ε>0

由于0<a<1,ε>0,

根据幂函数或指数函数的性质,得aε<1和(1+

ε
a
)a>1,

所以aε<(1+

ε
a
)aaaaεaa(1+
ε
a
)aaa+ε<(a+ε)a

即ab<ba.这与ab=ba矛盾,所以a不能小于b

假如b<a,则b<a<1,可设a=b+ε,其中ε>0,同上可证得ab<ba

这于ab=ba矛盾,所以a不能大于b

因此a=b

选择题
阅读理解

If you were on the street in Mexico today you might think you were in an operating room, surrounded by doctors. You could see many concerned people with masks (口罩) trying their best to stop the swine flu (猪流感). And the masks aren’t only being used in Mexico. A friend of mine traveling from New York City to Florida was given a mask and a pair of gloves just after he reached the airport. And another friend went to get a haircut only to be faced with a hairdresser wearing a mask!

But do the masks really work? Is covering your nose and mouth the key to stopping the swine flu? To find out the truth, ABC’s reporter Sharyn Alfonsi talked with germ (细菌) expert Dr. Elaine Larsen. It turns out that when you sneeze (打喷嚏) the air coming out of your mouth at a speed of one hundred miles per hour carries germs which can travel anywhere around you and make people around you infected (被感染).

However, whether you become infected actually depends on the weather. As Sharyn reported, when you sneeze, the germs leave your body in small drops of water coming out of your mouth. If the weather is wet, the wet air will make the small drops bigger and heavier and they drop down towards our feet. If this happens, we won’t become infected. But if the air is dry, those small drops can float higher up, making it possible for them to touch someone else’s nose, mouth or their mask.

Larsen says the masks can stop the germs well but after a few hours, they start to get wet, holding a lot of germs they come across and causing you to breathe them in. The masks do work, but the key to stopping the germs is changing your masks often.

小题1:According to Para.1, in Mexico you can see that ____.

A.the patients are treated in the open air

B.some doctors work in the street

C.people speak little to each other in the street

D.many people wear masks in the street小题2:According to the passage a patient spreads germs mainly through ____.

A.breathing

B.Speaking

C.touching others

D.sneezing小题3:We learn from the passage that on a wet day ____.

A.sick people will be infected

B.germs can spread very far

C.germs spread very quickly

D.few people will be infected小题4:If you keep wearing a mask for a long time, ____.

A.the mask will be old and worn out

B.you’ll love the feeling of wearing it

C.germs will be kept away from you

D.it will be easy for you to be infected小题5:What should we pay attention to when wearing masks?

A.Keeping the masks dry.

B.Changing the masks often.

C.Choosing very thick masks.

D.Wearing big masks.