0Î [a,b]. 3 : 1) 0=, 2) 0=b, 3)0Î(a,b). 0Î(a,b). 0 , f¢(x0), f¢(x0)=0. f¢(x0) .
f(x) , f¢(x) , .
, x0 f(x). , f(x) (a; b) {x1,x2, ,xn}. , x0, ( ) , : a,b,x1,xn. f(x) [a,b] fmax=max{f(a),f(b),f(x1),f(xn)}. fmin=min { f(a),f(b),f(x1),f(xn)}.
18. .
1. , . . .
(: ( ), →∞ f (x)/ ( ) f (x)- (b) ( =+b). .1.3.
2. , . .
(: f (-x)= f (x); f (-x)=- f (x). f (x+)= f (x)= f (x-))
3. . (, , . , . , ).
4., , . ( , , : -, +)
5. .
- = f(x) [a;b] . , = 0.
¹ a, ¹ b, f(c) max. , f'(c) = 0.
.. f(c) - max, f(x) £ f(c) xÎ[a;b]
f(x) - f(c) £ 0
.. - f , f'(c) = lim (f(x)-f(c))/(x-c)
1) x-c < 0 f(c)³ 0ü Þ f(c) = 0
2) x-c > 0 f(c)£ 0þ
, - .
1) - f(x) [a;b]; 2) , , (a;b); 3) - f(a) = f(b). a b , = 0.
|
|
-:
-, , - f(x) max min .
f(x1) = M max, f(x2) = m min; x1;x2 Î [a;b]
1) M = m, .. m £ f(x) £ M
Þ - f(x) a b , Þ . f(x)=0
2) M>m
.. f(a) = f(b) Þ - , Þ M m . f(c)=0.
1) - f(x) [a;b]
2) , , (a;b).
a b , : (f(b)-f(a))/(b-a)=f(c), a < c< b
-:
- F(x).
F(x) = f(x) - f(a) - [(f(b)-f(a))/(b-a)]*(x-a)
- :
1) ;
2) (a;b) -.
F(x) = f(x) - (f(b)-f(a))/(b-a)
3) a b - 0
F(a) = f(a) - f(a) - (f(b)-f(a))/(b-a)*( - ) = 0
F(b) = f(b) - f(a) - (f(b)-f(a))/(b-a)*(b-a) = 0
Þ - 0. F() = 0
f(c) - (f(b)-f(a))/(b-a) = 0,
f(c) = (f(b)-f(a))/(b-a)
CB/AC = (f(b)-f(a))/(b-a)
, || .