. f(x) [, b], f(a) = f(b) [, b] x0 (a < x0 < b), f ' (x0) = 0.
. .
1. f(x) [, b]; f ' (x) = 0 x (a < x < b), .. .
2. f(x) ( 1); , f(b) = f(a), f(a) - , f(x) .
.1 .2
f(x0) - f(x) [, b] x0 - . f(x0) : f(x0) f(x) x x0 [ , (, b)].
, , f(x) x0 , , f ' (x0) = 0, .
: AB y = f(x), , A B Ox, , t , Ox ( 1).
a, A B AB Ox', t AB ( 1). , :
AB y = f(x) , , AB ( 2).
, .
. f(x) [, b] f ' (x), x0 (a < x0 < b), f(b) - f(a) = (b - a)f '(x).
.
F(x) = f(x) - k(x - a),
- AB ( 2).
.
, x = a F(a) = f(a) - k(a - a) = f(a), x = b
, f(x) k(x - a) [a, b] (a, b), F(x) = f(x) - k(x - a) [a, b] (a, b).
, , (a, b) x0, F'(x0) = 0, .. f ' (x0) - k = 0
f(b) - f(a) = (b - a)f ' (x0),
.
a + (b - a) = b, a + Q(b - a), Q - (0 < Q < 1), - (a, b),
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f(b) - f(a) = (b - a)f ' [a + Q(b - a)]
a = x, b = x + Dx, b - a = Dx,
Dy = f(x + Dx) - f(x) = Dxf ' (x + QDx).
, C x (a, b), .
, :
f ' (x) x (a, b), f(x) = C.
, x1 x2 - (a, b), , f(x2) - f(x1) = (x2 - x1)f'(x0), , x1 < x0 < x2. f'(x0) = 0, f(x2) - f(x1) = 0, .
:
f1 (x) f2 (x) (a, b), .
, j(x) = f2(x) - f1(x). x (a, b) j'(x) = f2'(x) - f1'(x) = 0. , j(x) = C , f2(x) - f1(x) = .