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1、 Taylors Formula and the Study of Extrema 1.Taylors Formula for Mappings Theorem 1.If a mapping f:U Y from a neighborhood U U x of a point x in a normed space X into a normed space Y has derivatives up to order n1-inclusive in U and has an n-th order derivative f n x at the point x,then f x h f x f,
2、x h 1 f n xhn o hn ash 0.Equality(1)is one of the varieties of Taylors formula,written here for rather general classes of mappings.Proof.We prove Taylors formula by induction.For n 1 it is true by definition of f,x.Assume formula(1)is true for some n 1 N.Then by the mean-value theorem,formula(12)of
3、Sect.10.5,and the induction hypothesis,we obtain.ash 0.We shall not take the time here to discuss other versions of Taylors formula,which are sometimes quite useful.They were discussed earlier in detail for numerical functions.At this point we leave it to the reader to derive them(see,for example,Pr
4、oblem 1 below).2.Methods of Studying Interior Extrema Using Taylors formula,we shall exhibit necessary conditions and also sufficient conditions for an interior local extremum of real-valued functions defined on an open subset of a normed space.As we shall see,these conditions are analogous to the d
5、ifferential conditions already known to us for an extremum of a real-valued function of a real variable.Theorem 2.Let f:U R be a real-valued function defined on an open set U in a normed space X and having continuous derivatives up to order k 1 1 inclusive in a neighborhood of a point x U and a deri
6、vative f k x of order k at the point x itself.If f,x 0,f k 1 x 0 and f k x 0,then for x to be an extremum of the function f it is:necessary that k be even and that the formf k x hk be semidefinite,and sufficient that the values of the form f k x hk on the unit sphere h 1 be bounded away from zero;mo
7、reover,x is a local minimum if the inequalities f k xhk 0,(1)1 n 1!f n x hn 1 h f x h sup f,x h f,x f,x h 01 hold on that sphere,and a local maximum if f k xhk 0,Proof.For the proof we consider the Taylor expansion(1)of f in a neighborhood of x.The assumptions enable us to write f x h f x k1!f k x h
8、k h h k where h is a real-valued function,and h 0 ash 0.We first prove the necessary conditions.Since f k x 0,there exists a vector h0 0on which f k xh0k 0.Then for values of the real parametert sufficiently close to zero,x th0 f x k1!f k x th0 k th0 th0 k1!f k x h0k th0 h0 k tk and the expression i
9、n the outer parentheses has the same signf aksx h0k.For x to be an extremum it is necessary for the left-hand side(and hence also the right-hand side)of this last equality to be of constant sign when t changes sign.But this is possible only if k is even.This reasoning shows that ifx is an extremum,t
10、hen the sign of the difference f x th0 f x is the same as that of f k x h0k for sufficiently small t;hence in that case there cannot be two vectors h0,h1 at which the form f k x assumes values with opposite signs.We now turn to the proof of the sufficiency conditions.For definiteness we consider the
11、 case when f k x hk 0 for h 1.Then f x h f x k1!f k x hk h h and,since h 0 ash 0,the last term in this inequality is positive for all vectors h 0 sufficiently close to zero.Thus,for all such vectors h,f x h f x 0,that is,x is a strict local minimum.The sufficient condition for a strict local maximum
12、 is verified similiarly.Remark 1.If the space X is finite-dimensional,the unit sphere S x;1 with center atx X,being a closed bounded subset of X,is compact.Then the continuous function f k xhk i1 ikf xhi1 hik(a k-form)has both a maximal and a minimal value on Sx;1.If these values are of opposite sig
13、n,then f does not have an extremum atx.If they are both of the same sign,then,as was shown in Theorem 2,there is an extremum.In the latter case,a sufficient condition for an extremum can obviously be stated as the equivalent requirement that the form f k x hk be either positive-or negative-definite.
14、It was this form of the condition that we encountered in studying realvalued functions onRn.Remark 2.As we have seen in the example of functionsf:Rn R,the semi-definiteness of the 在内的导数而在点处有阶导数那么当时有等式是种形式的泰勒公式中的一种这一次它确实是对非常一般的函数类写出来的公式了我们用归纳法证明泰勒公式当时由的定义式成立假设对成立于是根据有限增量定理成立章中公式和所作时曾详细地讨论过它们现在我们把它们的结
15、论提供给读者例如可参看练习内部极值的研究我们将利用泰勒公式指出定义在赋范空间的开集上的实值函数在定义域内部取得局部极值的必要微分条件和充分微分条件我们将看到这些条件的邻域有直到阶包括阶在内的导映射在点本身有阶导映射如果且那么为使是函数的极值点必要条件是是偶数是半定的充分条件是在单位球面上的值不为零这时如果在这个球面上那么是严格局部极小点如果那么是严格局部极大点为了form f k x hk exhibited in the necessary conditions for an extremum is not a sufficient criterion for an extremum.Rem
16、ark 3.In practice,when studying extrema of differentiable functions one normally uses only the first or second differentials.If the uniqueness and type of extremum are obvious from the meaning of the problem being studied,one can restrict attention to the first differential when seeking an extremum,
17、simply finding the point x where f,x 0 3.Some Examples Example 1.Let L C1 R3;R and f C1 a,b;R.In other words,u1,u2,u 3 L u1,u2,u3 is a continuously differentiable real-valued function defined in R3 and x f x a smooth real-valued function defined on the closed interval a,b R.Consider the function F:C
18、 1 a,b;R R(2)defined by the relation f C 1 a,b;R F f b L x,f x,f,x dx R(3)a Thus,(2)is a real-valued functional defined on the set of functions C 1 a,b;R.The basic variational principles connected with motion are known in physics and mechanics.According to these principles,the actual motions are dis
19、tinguished among all the conceivable motions in that they proceed along trajectories along which certain functionals have an extremum.Questions connected with the extrema of functionals are central in optimal 在内的导数而在点处有阶导数那么当时有等式是种形式的泰勒公式中的一种这一次它确实是对非常一般的函数类写出来的公式了我们用归纳法证明泰勒公式当时由的定义式成立假设对成立于是根据有限增量定
20、理成立章中公式和所作时曾详细地讨论过它们现在我们把它们的结论提供给读者例如可参看练习内部极值的研究我们将利用泰勒公式指出定义在赋范空间的开集上的实值函数在定义域内部取得局部极值的必要微分条件和充分微分条件我们将看到这些条件的邻域有直到阶包括阶在内的导映射在点本身有阶导映射如果且那么为使是函数的极值点必要条件是是偶数是半定的充分条件是在单位球面上的值不为零这时如果在这个球面上那么是严格局部极小点如果那么是严格局部极大点为了(where f c is the maximum absolute value of the function on the closed interval a,b),the
21、n,settingu1 x,u2 f x,u3 f,x,1 0,2 hx,and3 h,x,we obtain from inequality(8),taking account of the uniform continuity of the functions iLu1,u2,u3,i 1,2,3,on boundedcontrol theory.Thus,finding and studying the extrema of functionals is a problem of intrinsic importance,and the theory associated with it
22、 is the subject of a large area of analysis-the calculus of variations.We have already done a few things to make the transition from the analysis of the extrema of numerical functions to the problem of finding and studying extrema of functionals seem natural to the reader.However,we shall not go dee
23、ply into the special problems of variational calculus,but rather use the example of the functional(3)to illustrate only the general ideas of differentiation and study of local extrema considered above.We shall show that the functional(3)is a differentiate mapping and find its differential.We remark
24、that the function(3)can be regarded as the composition of the mappings F1 f x Lx,f x,f,x(4)defined by the formula F1:C 1 a,b;R C a,b;R(5)followed by the mapping g C a,b;R F2 g g x dx R a(6)By properties of the integral,the mapping F2 is obviously linear and continuous,so that its differentiability i
25、s clear.We shall show that the mapping F1 is also differentiable,and that F1,f hx 2Lx,f x,f,xhx 3Lx,f x.f,xh,x(7)for h C 1 a,b;R.Indeed,by the corollary to the mean-value theorem,we can write in the present case L u1 1,u2 2,u3 3 L u1,u2,u3 sup 1Lu 1Lu1 2Lu 2Lu1 3Lu 3Lu 01 30max1 iL u u iL u im1a,2x,
26、3 i 1,2,3(8)where u u1,u2,u3 and 1,2,3.If we now recall that the norm f c 1 of the function f in C 1 a,b;R is 3 iL u1 i1 f max f c 在内的导数而在点处有阶导数那么当时有等式是种形式的泰勒公式中的一种这一次它确实是对非常一般的函数类写出来的公式了我们用归纳法证明泰勒公式当时由的定义式成立假设对成立于是根据有限增量定理成立章中公式和所作时曾详细地讨论过它们现在我们把它们的结论提供给读者例如可参看练习内部极值的研究我们将利用泰勒公式指出定义在赋范空间的开集上的实值函数在定
27、义域内部取得局部极值的必要微分条件和充分微分条件我们将看到这些条件的邻域有直到阶包括阶在内的导映射在点本身有阶导映射如果且那么为使是函数的极值点必要条件是是偶数是半定的充分条件是在单位球面上的值不为零这时如果在这个球面上那么是严格局部极小点如果那么是严格局部极大点为了subsets of R3,that maxLx,f x hx,f,x h,x Lx,f x,f,x 2Lx,f x,f,x hx 3Lx,f x,f,x h,x o h c1 as h c 1 0 But this means that Eq.(7)holds.By the chain rule for differentiat
28、ing a composite function,we now conclude that the functional(3)is indeed differentiable,and b F,f h 2Lx,f x,f,x hx 3Lx,f x,f,x h,x dx(9)a We often consider the restriction of the functional(3)to the affine space consisting of the functions f C 1 a,b;R that assume fixed values f a A,f b B at the endp
29、oints of the closed interval a,b.In this case,the functions h in the tangent spaceTC f1,must have the value zero at the endpoints of the closed interval a,b.Taking this fact into account,we may integrate by parts in(9)and bring it into the form 2Lx,f x,f,x ddx 3Lx,f x,f,x hx dx of course under the a
30、ssumption thaLt and f belong to the corresponding class C 2.In particular,if f is an extremum(extremal)of such a functional,then by Theorem 2 we have F,f h 0 for every function h C1 a,b;R such that ha h b 0.From this and relation(10)one can easily conclude(see Problem 3 below)that the function f mus
31、t satisfy the equation 2Lx,f x,f,x ddx 3Lx,f x,f,x 0(11)This is a frequently-encountered form of the equation known in the calculus of variations as the Euler-Lagrange equation.Let us now consider some specific examples.Example 2.The shortest-path problem Among all the curves in a plane joining two
32、fixed points,find the curve that has minimal length.The answer in this case is obvious,and it rather serves as a check on the formal computations we will be doing later.We shall assume that a fixed Cartesian coordinate system has been chosen in the plane,in which the two points are,for example,0,0 a
33、nd 1,0.We confine ourselves to just the curves that are the graphs of functions f C1 0,1;R assuming the value zero at both ends of the closed interval 0,1.The length of such a curve F f 1 f,x dx(12)depends on the function f and is a functional of the type considered in Example 1.In this case the fun
34、ction L has the form b F,f h(10)在内的导数而在点处有阶导数那么当时有等式是种形式的泰勒公式中的一种这一次它确实是对非常一般的函数类写出来的公式了我们用归纳法证明泰勒公式当时由的定义式成立假设对成立于是根据有限增量定理成立章中公式和所作时曾详细地讨论过它们现在我们把它们的结论提供给读者例如可参看练习内部极值的研究我们将利用泰勒公式指出定义在赋范空间的开集上的实值函数在定义域内部取得局部极值的必要微分条件和充分微分条件我们将看到这些条件的邻域有直到阶包括阶在内的导映射在点本身有阶导映射如果且那么为使是函数的极值点必要条件是是偶数是半定的充分条件是在单位球面上的值不为
35、零这时如果在这个球面上那么是严格局部极小点如果那么是严格局部极大点为了and therefore the necessary condition(11)for an extremal here reduces to the equation dx from which it follows that f x 常数 1 f,2 x on the closed interval 0,1 Since the function u is not constant on any interval,Eq.(13)is possible only if 2 f,x const on a,b.Thus a s
36、mooth extremal of this problem must be a linear function whose graph passes through the points 0,0 and 1,0.It follows that f x 0,and we arrive at the closed interval of the line joining the two given points.Example 3.The brachistochrone problem The classical brachistochrone problem,posed by Johann B
37、ernoulli I in 1696,was to find the shape of a track along which a point mass would pass from a prescribed point P0 to another fixed point P1 at a lower level under the action of gravity in the shortest time.We neglect friction,of course.In addition,we shall assume that the trivial case in which both
38、 points lie on the same vertical line is excluded.In the vertical plane passing through the points P0 and P1 we introduce a rectangular coordinate system such that P0 is at the origin,the x-axis is directed vertically downward,and the point P1 has positive coordinates x1,y1.We shall find the shape o
39、f the track among the graphs of smooth functions defined on the closed interval 0,x1 and satisfying the condition f 0 0,f x1 y1.At the moment we shall not take time to discuss this by no means uncontroversial assumption(see Problem 4 below).If the particle began its descent from the point P0 with ze
40、ro velocity,the law of variation of its velocity in these coordinates can be written as(14)Recalling that the differential of the arc length is computed by the formula we find the time of descent(13)dx 1 f,2 x ds 在内的导数而在点处有阶导数那么当时有等式是种形式的泰勒公式中的一种这一次它确实是对非常一般的函数类写出来的公式了我们用归纳法证明泰勒公式当时由的定义式成立假设对成立于是根据有
41、限增量定理成立章中公式和所作时曾详细地讨论过它们现在我们把它们的结论提供给读者例如可参看练习内部极值的研究我们将利用泰勒公式指出定义在赋范空间的开集上的实值函数在定义域内部取得局部极值的必要微分条件和充分微分条件我们将看到这些条件的邻域有直到阶包括阶在内的导映射在点本身有阶导映射如果且那么为使是函数的极值点必要条件是是偶数是半定的充分条件是在单位球面上的值不为零这时如果在这个球面上那么是严格局部极小点如果那么是严格局部极大点为了 along the trajectory defined by the graph of the function y f x on the closed inter
42、val0,x1.For the functional(16)L u1,u2,u3 and therefore the condition(11)for an extremum reduces in this case to the equation from which it follows that 1 f,2 x where c is a nonzero constant,since the points are not both on the same vertical line.Taking account of(15),we can rewrite(17)in the form dy
43、 c x ds However,from the geometric point of view dx cos,dy sin ds ds where is the angle between the tangent to the trajectory and the positive x-axis.By comparing Eq.(18)with the second equation in(19),we find 12 x 2 sin c2 But it follows from(19)and(20)that from which we find d dx f,x 0,(17)(18)(19
44、)(20)d sin2 d c2 c2 1 y 2c12 2 sin2 b(21)Setting 12 a and 2c2 2 t,we write relations(20)and(21)as x a 1 cost y a t sint b that x 0 only for t 2k,k Z.It follows from the form of the(22)Since a 0,it follows function(22)that we may assume without loss of generality that the parameter value t 0 correspo
45、nds to the point P0 0,0.In this case Eq.(21)implies b 0,and we arrive at the Ff,2 1 f,x dx x ddy ddyx,ddx tg ddx tg 2sin2 在内的导数而在点处有阶导数那么当时有等式是种形式的泰勒公式中的一种这一次它确实是对非常一般的函数类写出来的公式了我们用归纳法证明泰勒公式当时由的定义式成立假设对成立于是根据有限增量定理成立章中公式和所作时曾详细地讨论过它们现在我们把它们的结论提供给读者例如可参看练习内部极值的研究我们将利用泰勒公式指出定义在赋范空间的开集上的实值函数在定义域内部取得局部极
46、值的必要微分条件和充分微分条件我们将看到这些条件的邻域有直到阶包括阶在内的导映射在点本身有阶导映射如果且那么为使是函数的极值点必要条件是是偶数是半定的充分条件是在单位球面上的值不为零这时如果在这个球面上那么是严格局部极小点如果那么是严格局部极大点为了 x a 1 cost y a t sint for the parametric definition of this curve.Thus the brachistochrone is a cycloid having a cusp at the initial point P0 where the tangent is vertical.Th
47、e constant a,which is a scaling coefficient,must be chosen so that the curve(23)also passes through the point P1.Such a choice,as one can see by sketching the curve(23),is by no means always unique,and this shows that the necessary condition(11)for an extremum is in general not sufficient.However,fr
48、om physical considerations it is clear which of the possible values of the parametera should be preferred(and this,of course,can be confirmed by direct computation).simpler form(23)在内的导数而在点处有阶导数那么当时有等式是种形式的泰勒公式中的一种这一次它确实是对非常一般的函数类写出来的公式了我们用归纳法证明泰勒公式当时由的定义式成立假设对成立于是根据有限增量定理成立章中公式和所作时曾详细地讨论过它们现在我们把它们的
49、结论提供给读者例如可参看练习内部极值的研究我们将利用泰勒公式指出定义在赋范空间的开集上的实值函数在定义域内部取得局部极值的必要微分条件和充分微分条件我们将看到这些条件的邻域有直到阶包括阶在内的导映射在点本身有阶导映射如果且那么为使是函数的极值点必要条件是是偶数是半定的充分条件是在单位球面上的值不为零这时如果在这个球面上那么是严格局部极小点如果那么是严格局部极大点为了泰勒公式和极值的研究 这里我们不再继续讨论其他的，有时甚至是十分有用的泰勒公式形式。当时，在研 究数值函数时，曾详细地讨论过它们。现在我们把它们的结论提供给读者(例如，可参 看练习 1)。2.内部极值的研究 我们将利用泰勒公式指出定
50、义在赋范空间的开集上的实值函数在定义域内部取得 局部极值的必要微分条件和充分微分条件。我们将看到，这些条件类似于我们熟知的实 变量的实值函数的极值的微分条件。定理2 设f:U R是定义在赋范空间 X的开集 U上的实值函数，且f 在某个点 x U 的邻域有直到 k 1 1阶.(包括k-1 阶在内的)导映射，在点 x本身有 k阶导映射 fk x 如果 f,x 0,f k1 x 0且f k x 0，那么为使 x是函数 f 的极值点 必要条件是:k 是偶数，f k x hk 是半定的。充分条件是:f k x hk在单位球面 h 1上的值不为零；这时，如果在这个球面上1.映射的泰勒公式 定理 1 如果从