Конспект лекций по математическому анализу. Шерстнев А.Н. - 452 стр.

BUDEM BRATX WEKTORY g WIDA       th (t ! 0; t > 0). tOGDA DLQ DOSTATO^NO
              j o( th ) j  0(f )h
MALYH t > 0 : kthk < 2kh0k . dLQ TAKIH t POLU^IM
                                                    0
                                            f0)th + o(th) ]
            (f0 + th) , (f0) = kthk[  k(th  k kthk
                                        0 (f )h o(th)
                               = kthk[ kh0k + kthk ]
                                kthk[  k(hf0k)h , jok(ththk)j ] > 0;
                                         0


| PROTIWORE^IE S LOKALXNYM MAKSIMUMOM W TO^KE f0: >
   3. p R I M E R. wERN  EMSQ K PRIMERU 256.8. eSLI NA[ FUNKCIONAL
       Zb
(x) = f (t; x(t)) dt OBLADAET LOKALXNYM \KSTREMUMOM W TO^KE x0, TO
        a
                         Zb
             0(x0)h =        fv0 (t; x0(t))h(t) dt = 0 (h 2 C [a; b]):
                          a
oTS@DA SLEDUET, ^TO fv0 (t; x0(t)) = 0 (!!).
   x258. oCENO^NAQ FORMULA lAGRANVA
   pUSTX E; F | NORMIROWANNYE PROSTRANSTWA NAD POLEM (= C ILI
R); U ( E ) | OTKRYTO, OTREZOK [x; x + h] = fx + th j 0  t  1g
SODERVITSQ W U I OTOBRAVENIE A : U ! F DIFFERENCIRUEMO NA \TOM
OTREZKE. tOGDA
              kA(x + h) , A(x)k  sup kA0(x + h)k khk:
                                          2[0;1]

 pUSTX FUNKCIONAL ' 2 F  PROIZWOLEN. pOLOVIM f (t)  '(A(x+th)) (0 
t  1). |TA ^ISLOWAQ FUNKCIQ ^ISLOWOGO ARGUMENTA DIFFERENCIRUEMA PO
t NA INTERWALE (0; 1) W SILU 256.7, PRI^EM f 0(t) = '(A0(x + th)h) (0 <
t < 1). pRIMENQQ K f FORMULU KONE^NYH PRIRA]ENIJ lAGRANVA, IMEEM
f (1) , f (0) = f 0() (0 <  < 1);  = ('), TO ESTX
   j'(A(x + h) , A(x))j = j'(A0(x + h)h)j  k'k sup kA0(x + h)k khk:
                                                        01


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