Конспект лекций по математическому анализу. Шерстнев А.Н. - 60 стр.

NAJDETSQ c 2 (a; b) TAKOE, ^TO f (c) = sup f (x). pRI \TOM
                                      x2[a;b]

                  f 0(c+) = hlim f (c + h) , f (c)  0;
                             !0+          h
                  f (c,) = hlim
                    0            f ( c + h ) , f (c)  0:
                             !0,          h
sLEDOWATELXNO, f 0(c) = f 0(c+) = f 0(c,) = 0 (SM. 29.12). >
    2. t E O R E M A [o. kO[I]. pUSTX f; g : [a; b] ! R NEPRERYWNY I NA
(a; b) DIFFERENCIRUEMY, PRI^EM f 0(x); g0(x) NE RAWNY NUL@ ODNOWREMENNO
I g(b) =6 g(a). tOGDA SU]ESTWUET c (a < c < b) TAKOE, ^TO
                              f (b) , f (a) = f 0(c) :
                               g(b) , g(a) g0(c)
  fUNKCIQ h(x) = g(x)[f (b) , f (a)] , f (x)[g(b) , g(a)] UDOWLETWORQET
USLOWIQM TEOREMY rOLLQ. pO\TOMU SU]ESTWUET c 2 (a; b) TAKOE, ^TO
              h0(c) = g0(c)[f (b) , f (a)] , f 0(c)[g(b) , g(a)] = 0:
zAMETIM, ^TO g0(c) = 6 0, IBO INA^E f 0(c) = 0; ^TO PROTIWORE^IT PREDPOLO-
VENI@ TEOREMY. oTS@DA SLEDUET ISKOMOE RAWENSTWO. >
    3. [fORMULA lAGRANVA (KONE^NYH PRIRA]ENIJ)]. pUSTX f : [a; b] ! R
NEPRERYWNA I NA (a; b) DIFFERENCIRUEMA. tOGDA SU]ESTWUET c (a < c < b)
TAKOE, ^TO f (b) , f (a) = f 0(c)(b , a).
  pOLOVIM W TEOREME kO[I g(x) = x (a  x  b): >
    4. s L E D S T W I E. pUSTX f : [a; b] ! R NEPRERYWNA I NA (a; b)
DIFFERENCIRUEMA, PRI^EM f 0(x) = 0 (a < x < b). tOGDA f = const.
  dLQ L@BOGO x 2 (a; b] : f (x) , f (a) = f 0(c)(x , a) = 0: >
    z A M E ^ A N I Q. 5. w USLOWIQH FORMULY lAGRANVA DLQ L@BOGO
x 2 (a; b)
()        f (x + h) , f (x) = f 0(x + h)h (0 <  < 1;  = (h)):
|TO SOOTNO[ENIE POLEZNO SOPOSTAWITX S RAWENSTWOM
                 f (x + h) , f (x) = f 0(x)h + o(h) (h ! 0):
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