admin 管理员组

文章数量: 1086019


2024年3月26日发(作者：数字推盘有无解的情况吗)

外文原文

EXTREME VALUES OF FUNCTIONS OF SEVERAL

REAL VARIABLES

1. Stationary Points

Definition 1.1 Let

DR

n

and

f:DR

. The point a

aD

is said to be:

(1) a local maximum if

f(x)f(a)

for all points

x

sufficiently close to

a

;

(2) a local minimum if

f(x)f(a)

for all points

x

sufficiently close to

a

;

(3) a global (or absolute) maximum if

f(x)f(a)

for all points

xD

;

(4) a global (or absolute) minimum if

f(x)f(a)

for all points

xD

;;

(5) a local or global extremum if it is a local or global maximum or minimum.

Definition 1.2 Let

DR

n

and

f:DR

. The point a

aD

is said to be

critical or stationary point if

f(a)0

and a singular point if

f

does not exist

at

a

.

Fact 1.3 Let

DR

n

and

f:DR

.If

f

has a local or global extremum at the

point

aD

, then

a

must be either:

(1) a critical point of

f

, or

(2) a singular point of

f

, or

(3) a boundary point of

D

.

Fact 1.4 If

f

is a continuous function on a closed bounded set then

f

is bounded

and attains its bounds.

Definition 1.5 A critical point

a

which is neither a local maximum nor minimum is

called a saddle point.

Fact 1.6 A critical point

a

is a saddle point if and only if there are arbitrarily small

values of

h

for which

f(ah)f(a)

takes both positive and negative values.

Definition 1.7 If

f:R

2

R

is a function of two variables such that all second

order partial derivatives exist at the point

(a,b)

, then the Hessian matrix of

f

at

(a,b)

is the matrix

f

xx

H

f

yx

where the derivatives are evaluated at

(a,b)

.

f

xy

f

yy

If

f:R

3

R

is a function of three variables such that all second order partial

derivatives exist at the point

(a,b,c)

, then the Hessian of f at

(a,b,c)

is the matrix

f

xx

H

f

yx

f

zx

f

xy

f

yy

f

zy

f

xz

f

yz

f

zz

where the derivatives are evaluated at

(a,b,c)

.

Definition 1.8 Let

A

be an

nn

matrix and, for each

1rn

,let

A

r

be the

rr

matrix formed from the first

r

rows and

r

columns of

A

.The determinants

det(

A

r

),

1rn

,are called the leading minors of

A

Theorem 1.9(The Leading Minor Test). Suppose that

f:R

2

R

is a sufficiently

smooth function of two variables with a critical point at

(a,b)

and H the Hessian of

f

at

(a,b)

.If

det(H)0

, then

(a,b)

is:

2

(1) a local maximum if 0>det(H

1

) = f

xx

and 0

f

xx

f

yy

f

xy

;

2

(2) a local minimum if 0

1

) = f

xx

and 0

f

xx

f

yy

f

xy

;

(3) a saddle point if neither of the above hold.

where the partial derivatives are evaluated at

(a,b)

.

Suppose that

f:R

3

R

is a sufficiently smooth function of three variables with a

critical point at

(a,b,c)

and Hessian H at

(a,b,c)

.If

det(H)0

, then

(a,b,c)

is:


本文标签： 有无 推盘 外文 原文 数字

版权声明：本文标题：数学及应用数学专业外文翻译--多元函数的极值 内容由网友自发贡献，该文观点仅代表作者本人， 转载请联系作者并注明出处：http://roclinux.cn/b/1711411729a592732.html， 本站仅提供信息存储空间服务，不拥有所有权，不承担相关法律责任。如发现本站有涉嫌抄袭侵权/违法违规的内容，一经查实，本站将立刻删除。