, : .
n- :
, .
. .
20. , : . . .
. - y=f(x) [a,b] (a,b) - , .. f(a)=f(b), (a,b) .ξ, -=0. - -, : [a,b] - max min : 1) fmax=fmin, f(x)=const. 2) fmax>fmin, - .ξϵ(a,b). f(ξ)= fmax, ∆>0 - f(x+∆x)-f(x)<=0, f(x-∆x)-f(x)<=0, , ∆x , : (f(x+∆x)-f(x))/ ∆x<=0, (f(x-∆x)-f(x))/ -∆x>=0. - x=ξ: f ′(ξ)<=0, f ′(ξ)>=0, => f ′(ξ)=0.
. - f(x) φ(x) [a,b], (a,b) φ′(x)=0 (a,b), ξ , (f(b)-f(a))/ (φ(b)-φ(a))=f ′(ξ)/φ′(ξ) (1).
-: - F(x)=f(x)+λφ(x), λ , f(a)=f(b), .. F(x) .. λ, . : f()+λφ()= f(b)+λφ(b). λ= (f(b)- f())/(φ(b)-φ()). -:
F(x)=f(x)+ (f(b)- f())/(φ(b)-φ())φ(x)=0. φ(ξ)≠0 (f(b)- f())/(φ(b)-φ())= f ′(ξ)/φ′(ξ).
. - f(x) [a,b] (a,b), .ξϵ(a,b) , f(b)-f(a)=f ′(ξ)(b-a) (2).
-: - (2) (1). (1) φ(x)=x, φ′(x)=x′=1, φ(a)=a, φ(b)=b. - : (f(b)-f(a))/(b-a)= f ′(ξ), (2).
:
-. .
y=f(x), [ a, b ] ( (a, b)), , x [ a, b ] , x 1 < x 2, f(x 1 ) < f(x 2 ).
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y=f(x) [ a, b ], x [ a, b ] , x 1 < x 2, f(x 1 ) > f(x 2 ).
, , .
y=f(x) [ a, b ], x .
1*. ( )
y=f(x) [ a, b ], , f '(x) ≥ 0.
. y=f(x) [ a, b ], (a, b) , f ' (x) ≥ 0 a<x<b, f(x) [ a, b ].
* 0 =ƒ(), d - 0, ≠0 ƒ()<ƒ(0).
: x0 , $d>0 " : 0<|x-x0|<d Þ ƒ()>ƒ(0). () () . () .
, . .
*( ). =ƒ() d - 0 ( ) ƒ'() , 0 ; , 0 .
* :
1) =ƒ();
2) , ;
3) ƒ'() ;
4) ( ) ( ) .
* .
=ƒ() (;b), . =ƒ() (;b), .
=ƒ() (;b) , . . ƒ"()<0, . ƒ"()>0 " x(;b) .
=ƒ(), , .
( ). ƒ"() 0, , , 0 .
ƒ"()<0 <0 ƒ"()>0 >0. , =0 , . , (0;ƒ(0)) .
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, ƒ"()>0 <x0 ƒ"()<0 >0, (0;ƒ(0)) =ƒ().
, , , .
, .
, =
, , ƒ () . .
47(22)
,
-- , , -- , -- . , - , .
, , ,
.
49(24)
y = f(x) (a; b), . y = f(x) (a; b), . , (a; b) (b; c). , , . . y = f(x) (a; b). (a; b) y = f(x) , .. f ''(x) < 0, , f ''(x) > 0 . . , f ''(x) < 0 , . y = f(x) M0 x0 Î (a; b) M0 . . , (a; b) , .. x y = f(x) . |
, y = f(x). , x. . , x .
f(x) f(x0) , c x x0.
, .
, : , c1 c0 x0. f ''(x) < 0. .
1. , x > x 0. x0 < c1 < c < x, , (x x 0) > 0 (c x 0) > 0. .
2. x < x0, , x < c < c 1 < x 0 (x x 0) < 0, (c x 0) < 0. .
, x x0 Î (a; b), , . .
() .
f (x) ( ) (a, b), :
f '' (x) > 0 x (a, b), f (x) (a, b);
f '' (x) < 0 x (a, b), f (x) (a, b).
23. .
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