泰勒公式外文翻译

Taylor's Formula and the Study of Extrema

1. Taylor's Formula for Mappings

Theorem 1. If a mapping Y U f →: from a neighborhood ()x U U = of a point x in a normed space X into a normed space Y has derivatives up to order n -1 inclusive in U and has an n-th order derivative ()()x f n at the point x, then

()()()()()⎪⎭⎫ ⎝⎛++++=+n n n h o h x f n h x f x f h x f !1,Λ (1)

as 0→h .

Equality (1) is one of the varieties of Taylor's formula, written here for rather general classes of mappings.

Proof. We prove Taylor's formula by induction.

For 1=n it is true by definition of ()x f ,.

Assume formula (1) is true for some N n ∈-1.

Then by the mean-value theorem, formula (12) of Sect. 10.5, and the induction hypothesis, we obtain.

()()()()()()()()()()()()()⎪⎭⎫ ⎝⎛=⎪⎭⎫ ⎝⎛=⎪⎪⎭

⎫ ⎝⎛-+++-+≤⎪⎭

⎫ ⎝⎛+++-+--<

as 0→h .

We shall not take the time here to discuss other versions of Taylor's 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, Problem 1 below).

2. Methods of Studying Interior Extrema

Using Taylor's 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 differential conditions already known to us for an extremum of a real-valued function of a real variable.

Theorem 2. Let R U f →: be a real-valued function defined on an open set U in a normed space X and having continuous derivatives up to order 11≥-k inclusive in a neighborhood of a point

U x ∈ and a derivative ()()x f k of order k at the point x itself. If ()()()0,,01,==-x f x f k Λ and ()()0≠x f k , then for x to be an extremum of the function f it is:

necessary that k be even and that the form

()()k k h x f be semidefinite, and

sufficient that the values of the form ()()k k h x f on the unit sphere 1=h be bounded away from zero; moreover, x is a local minimum if the inequalities

()()0>≥δk k h x f ,

hold on that sphere, and a local maximum if

()()0<≤δk k h x f ,

Proof. For the proof we consider the Taylor expansion (1) of f in a neighborhood of x. The assumptions enable us to write

()()()()()k k k h h h x f k x f h x f α+=-+!1

where ()h α is a real-valued function, and ()0→h α as 0→h .

We first prove the necessary conditions.

Since ()()0≠x f k , there exists a vector 00≠h on which ()()00≠k k h x f . Then for values of the real parameter t sufficiently close to zero,

()()()()()()k k k th th th x f k x f th x f 0000!1α+=-+

()()()k

k k k t h th h x f k ⎪⎭⎫ ⎝⎛+=000!1α

and the expression in the outer parentheses has the same sign as

()()k k h x f 0. 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 if x is an extremum, then the sign of the difference ()()x f th x f -+0 is the same as that of

()()k k h x f 0 for sufficiently small t; hence in that case there cannot be two vectors 0h , 1h at which the form ()()x f k assumes values with opposite signs.

We now turn to the proof of the sufficiency conditions. For definiteness we consider the case when ()()0>≥δk k h x f for 1=h . Then

()()()()()k k k h h h x f k x f h x f α+=-+!1

()()()k k k h h h h x f k ⎪⎪⎪⎭

⎫ ⎝⎛+⎪⎪⎭⎫ ⎝⎛=α!1 ()k h h k ⎪⎭⎫ ⎝⎛+≥αδ!1

and, since ()0→h α as 0→h , the last term in this inequality is positive for all vectors 0≠h sufficiently close to zero. Thus, for all such vectors h,