问题 填空题
若函数f(x)定义域内有两个任意实数x1,x2,满足f(
x1+x2
2
)<
f(x1)+f(x2)
2
,则称函数f(x)为凸函数,下列函数中是凸函数的为 ______.
①f(x)=3x+1,②f(x)=
1
x
x∈(-∞,0),③f(x)=x2-3x-2,④f(x)=-|x+1|
答案

f(x1)+f(x2)
2
-f(
x1+x2
2
)=
3x1+1+3x2+1
2
-3
x1+x2
2
+1
=0,不是凸函数;

f(x1)+f(x2)
2
-f(
x1+x2
2
)=
1
x1
+
1
x2
2
-
1
x1 +x2
2
=
2(x1-x22
(x1 +x2x1x2
符号不确定,故不为凸函数

f(x1)+f(x2)
2
-f(
x1+x2
2
)=
x12-3x1-2+x22-3x2-2
2
-(
x1+x2
2
)
2
+3
x1+x2
2
+2

=(

x1-x2
2
)2>0,故为凸函数.

f(x1)+f(x2)
2
-f(
x1+x2
2
)=
-|x1+1|-|x2+1|
2
+|
x1 +x2
2
+1|

取x1=1,x2=2则上式为0,故不是凸函数.

故答案为:③

完形填空

If you often have a cold, headache or backache, there is something wrong with your health. Eating healthy food  good for our health. .Tofu(豆腐), milk, vegetables are healthy for us.

Tofu is a kind of traditional(传统的)   2  food. We often   3  it in China. Now more and more people   4  other countries think tofu is good for health,  5 . They also like eating it.

Milk is   6  kind of health food. Every day you should   7  one or two glasses of milk. It can   8  us strong.

Vegetables are very important. You should eat   9  them. There are vitamin(维他命) A, B, C, D , E and so on in them.

As we know, it’s important   10  a balanced diet(均衡饮食). We must have the right kinds of food. We should also eat some pork, chicken and beef. We should eat more fruit and vegetables but less meat.

(       )1 A are       B is         C will be    D was

(       )2 A Japanese  B American  C English   D Chinese

(       )3 A eat       B drink      C take     D play

(       )4 A for       B with       C in       D at

(       )5 A too       B also       C well   D either

(       )6 A the other  B other      C another  D others

(       )7 A drinking   B drink     C take     D to take

(       )8 A keep      B make    C keeps    D keeping

(       )9 A a lot       B plenty    C much    D  lots of

(       )10 A making   B to keep    C keeping   D to drink

单项选择题