外文原文
EXTREME VALUES OF FUNCTIONS OF SEVERAL
REAL VARIABLES
1. Stationary Points
Definition 1.1 Let n R D ⊆ and R D f →:. The point a D a ∈ is said to be:
(1) a local maximum  if )()(a f x f ≤for all points x  sufficiently close to a ;
(2) a local minimum if )()(a f x f ≥for all points x  sufficiently close to a ;
(3) a global (or absolute) maximum if )()(a f x f ≤for all points D x ∈;
(4) a global (or absolute) minimum if )()(a f x f ≥for all points D x ∈;;
(5) a local or global extremum  if it is a local or global maximum or minimum. Definition 1.2 Let n R D ⊆ and R D f →:. The point a D a ∈ is said to be critical or  stationary point  if 0)(=∇a f  and a singular point  if f ∇ does not exist at a .
Fact 1.3 Let n R D ⊆ and R D f →:.If f  has a local or global extremum at the point D a ∈, 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 )()(a f h a f -+ takes both positive and negative values.
Definition 1.7 If R R f →2: is a function of two variables such that all second order partial derivatives exist at the point ),(b a , then the Hessian matrix of f  at ),(b a  is the matrix
⎪⎪⎭
⎫  ⎝⎛=yy yx
xy xx f f f f H  where the derivatives are evaluated at ),(b a . If R R f →3: is a function of three variables such that all second order partial derivatives exist at the point ),,(c b a , then the Hessian of f at ),,(c b a  is the matrix
⎪⎪⎪⎭
⎫  ⎝⎛=zz zy zx yz yy yx xz xy xx f f f f f f f f f H  where the derivatives are evaluated at ),,(c b a .
Definition 1.8 Let A  be an n n ⨯ matrix and, for each n r ≤≤1,let r A be the r r ⨯ matrix formed from the first r  rows and r  columns of A .The determinants det(r A ),n r ≤≤1,are called the leading minors of A
Theorem 1.9(The Leading Minor Test). Suppose that R R f →2:is a sufficiently smooth function of two variables with a critical point at ),(b a and  H the Hessian  of f at ),(b a .If 0)det(≠H , then ),(b a  is:
(1) a local maximum if 0>det(H 1) = f xx  and  0<det(H )=2xy yy xx f f f -;
(2) a local minimum if 0<det(H 1) = f xx  and  0<det(H )=2xy yy xx f f f -;
(3) a saddle point if neither of the above hold.
where the partial derivatives are evaluated at ),(b a .
Suppose that R R f →3: is a sufficiently smooth function of three variables with a critical point at ),,(c b a and  Hessian H at ),,(c b a .If 0)det(≠H , then ),,(c b a  is:
(1) a local maximum if 0>det(H 1), 0<det(H 2) and 0>det(H 3);values翻译
(2) a local minimum if 0<det(H 1), 0<det(H 2) and 0>det(H 3);
(3) a saddle point if neither of the above hold.
where the partial derivatives are evaluated at ),,(c b a .
In each case, if det(H )= 0, then ),(b a  can be either a local extremum or a saddle
Example. Find and classify the stationary points of the following functions:
(1) ;1),,(2224+++++=xz z y y x x z y x f
(2) ;)1()1(),(422++++=x y x y y x f
Solution. (1) 1),,(2224+++++=xz z y y x x z y x f ,so
)24),(3z xy x y x f ++=∇（i )2(2y x ++j )2(x z ++k
Critical points occur when 0=∇f ,i.e. when
(1)                    z xy x ++=2403
(2)                    y x 202+=
(3)                    x z +=20
Using equations (2) and (3) to eliminate y and z from (1), we see that 02
1433=--x x x or 0)16(2=-x x ,giving 0=x ,66=x  and 66-=x .Hence we have three stationary points: ）（0,0,0,
）（126,121,66-- and ）（126,121,66--.    Since y x f xx 2122+=,x f xy 2=,1=xz f ,2=yy f ,0=yz f  and 2=zz f ,the Hessian matrix is
⎪⎪⎪⎭
⎫  ⎝⎛+=201022122122x x y x H    At ）（12
6,121,66--, ⎪⎪⎪⎪⎭
⎫    ⎝
⎛=201023/613/66/11H  which has leading minors 611>0, 039
631123/63/66/11det >=-=⎪⎪⎭⎫  ⎝⎛ And det 042912322>=--=
H .By the Leading Minor Test, then, ）（12
6,121,66--is a local minimum.    At ）（12
6,121,66--, ⎪⎪⎪⎪⎭
⎫    ⎝⎛--=201023/613/66/11H  which has leading minors 6
11>0,
039
631123/63/66/11det >=-=⎪⎪⎭⎫  ⎝⎛ And det 042912322>=--=
H .By the Leading Minor Test, then, ）（12
6,121,66--is also a local minimum. At ）（0,0,0, the Hessian is
⎪⎪⎪⎭
⎫  ⎝⎛=201020100H
Since det 2)(-=H , we can apply the leading minor test which tells us that this is a saddle point since the first leading minor is 0. An alternative method is as follows. In this case we consider the value of the expression
hl l k k h h l k h f f D ++++=+++-=22240,0,00,0,0）（）（，
for arbitrarily small values of h, k and l. But for very small h, k and l , cubic terms and above are negligible in comparison to quadratic and linear terms, so
that hl l k D ++≈22.If h, k and l  are all positive, 0>D . However, if 0=k  and
0<h  and h l <<0,then 0<D .Hence close to ）
（0,0,0,f both increases and decreases, so ）（0,0,0 is a saddle point.
(2) 422)1()1(),(++++=x y x y y x f so
))1(4)1(2(),(3+++=∇x y x y x f i ))1(2(2+++x y j .
Stationary points occur when 0=∇f ,i.e. at )0,1(-.
Let us classify this stationary point without considering the Leading Minor Test (in this case the Hessian has determinant 0 at )0,1(- so the test is not applicable). Let
.0,10,1422h k h k k h f f D ++=++---=）（）（

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