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4 âle _ofekoes expærflektmces

_ærfe 0 Cflftes et oaktfkuftæ

Oarrfiæ he c‖exerofoe =

Ak m 0

( x + = ∔ = )( x + = + = )

x + = ∔ = x + =

cfl < cfl < cfl

x ↔ > x ↔ > x ↔>

+ ∔

( )

tmk x tmk x x = =

Hako cfl j ( x ) < j ( > )

x ↔>

^mr sufte cm jakotfak j est oaktfkue kue ek >

∔ =

x + +

( )

tmk x = =

cfl

x ↔>

= = =

< Ù <

tmk x x + = + = 4

x

Oarrfiæ he c‖exerofoe 4

4

x + x ∔

=4

j x x

x ∔ ?

Ak m ( )

Hako ( )

< < +

8

paur taut x ≮ ?

4

x + x ∔

=4

cfl j x < cfl < cfl x + 8 < 9

x ↔ ? x ↔ ? x ∔ ? x ↔?

H‖aö cfl j ( x ) < j (?)

x ↔?

^mr sufte cm jakotfak j est oaktfkue kue ek ?

Ak m j ( )

( )

4 4 + =

4 < < 5

∔

Ak m 0 j ( x )

Et ( )

( )

9 ? 4

4 x + =

cfl < cfl < 5

9 ∔ ? x

x ↔ 4 x ↔4

x 34 x 34

Oarrfiæ he c‖exerofoe ?

( x ∔ 4 )( x + ?)

4

x + x ∔ :

cfl j x < cfl < cfl < cfl x + ? < 5

x ∔ 4 x ∔ 4

x ↔ 4 x ↔ 4 x ↔ 4 x ↔4

x ; 4 x ; 4 x ; 4 x ; 4

^ufsque cfl j ( x ) cflj ( x ) j ( 4)

x ↔ 4 x ↔4

x ; 4 x 34

< < mcars j est oaktfkue ek 4

---

4 âle _ofekoes expærflektmces

_ærfe 0 Cflftes et oaktfkuftæ

Oarrfiæ he c‖exerofoe 8

Ak m 0

cfl j

( x )

<

x ↔ = x ↔=

b ↔>

sfk

cfl

x

b ↔>

( ώ x )

=

b < x ∔=

( ώ ( + b ))

x

b

b >

↔

( ώ + ώ b )

sfk =

cfl =

sfk

cfl

b

( ώ b )

∔

sfk

cfl

b ↔>

b

t < ώ b

sfk ( ώ b )

cfl ώ

b >

↔

b >

ώ b t ↔ >

ώ

∔

< ↔

<

<

< ∔ ↔

< ∔

est oaktfkue ek = ⇘ cfl j ( x ) < j ( = )

j est oaktfkue ek

x ↔=

⇘ l < ∔ ώ

• Ak m

• Ak m

cfl

( x )

x ↔ > x ↔>

Hako

Hako

cfl

Oarrfiæ he c‖exerofoe 5

tmk

tmk x

ώ

< cfl Ù < et cm jakotfak "oas" est oaktfkue kue ek

? x ? x ?

?

ώ ώ ώ

cfloas

x ↔>

ώ tmk( x ) ώ

< oas <

x

tmk =

? ? 4

x 8 x ? x

cfl

x

x

4 4

ώ ώ ώ

∔ +

< <

x

4 4

↔+∛

8 + 9 x ↔+∛

8 8

4

ώ x ∔ 8 x + ? ώ 4

cfl sfk sfk

4 < <

↔+∛

8x

+ 9 8 4

x

ώ

et cm jakotfak "sfk"est oaktfkue ek

8

---

• Ak m

Hako

4

4x

4 4

cfl < cfl < < 8

x ↔ > = ∔ oas x x ↔>

= ∔ oas x =

4

x 4

4

4x

cfl < 8 < 4

x ↔>

= oas x

∔

et cm

4 âle _ofekoes expærflektmces

_ærfe 0 Cflftes et oaktfkuftæ

cm jako

kotfak " "est

oaktfkue

ek 8

( )

?

j x x x

< + ∔

=

=) Laktraks que c‖æqumtfak

Laktraks que c‖æqumtfak ( ) >

Ak m 0

Oarrfiæ he c‖exerofoe :

j x < mhlet uke sacutfak ukfque ν sur R

j est oaktfkue sur R ( omr j est uk pacykøle )

( R) ‴( )

x j x x j

4

∎ ∂ < ? + = ; > ⇔

( )

> ∂ j R ( omr

Hako c‖æqumtfak ( ) >

4) Ak m j est oaktfkue sur

est strfotelekt orafssmkte sur R

( omr j ( ) < j (\ ∔ , W) cfl j ( x ), cfl j ( x ) \ , W

est oaktfkue sur W >,= \

R < ∔∛ +∛ < < ∔∛ +∛ < R )

x ↔∔∛ x ↔+∛

j x < mhlet uke sacutfak ukfque ν sur R

>,= et

j

j

( > ) < ∔ =

⇔ j ( ) Ù j ( ) 3

( = ) =

<

> = >

Hako h‖mprâs ce tbæarâle hes vmce

ceurs fkterlæh

æhfmfres

0 > 3 ν 3 =

?) Etuhfaks ce sfike he

Etuhfaks ce sfike he j ( )

= er oms 0 sf x

4 âle oms 0 sf x

≨ ν

x sur R

Mcars j ( x ) j ( ν )

≨ ( omr j est orafssmkte sur R )

Hako j ( x ) ≨ > ( omr j ( ν ) < > )

≥ ν

Mcars j ( x ) j ( ν )

≥ ( omr j est orafssmkte sur R )

Hako j ( x ) ≥ > ( omr j ( ν ) < > )

---

4 âle _ofekoes expærflektmces

_ærfe 0 Cflftes et oaktfkuftæ

=) Laktraks que c‖æqumtfak

4)

F

Laktraks que c‖æqumtfak ( ) 0= sfk

Oarrfiæ he c‖exerofoe 9

E + x < x

mhlet mu lafks uke sacutfak sur

ώ 4ώ

c‖fktervmcce F <

,

4 ?

Oaksfhæraks cm jakotfak b hæjfkfe pmr 0 b x < + x ∔ x

Ak m 0

b est oaktfkue sur

F

hæjfkfe pmr 0 ( ) = sfk( )

ώ 4ώ

<

,

4 ?

ώ 8 ∔ ώ

b < ; >

4 4 ώ 4ώ

b b

4 ώ : + ? ? ∔ 8ώ

4 ?

b

>

< 3

? :

⇔ Ù 3

Hako h‖mprâs ce tbæarâle hes vmceurs fkterlæhfmfres 0 c‖æqumtfak

ώ 4ώ

lafks uke sacutfak sur c‖fktervmcce F <

,

4 ?

Hako h‖mprâs ce tbæarâle hes vmceurs fkterlæhfmfres 0 c‖æqumtfak ( ) >

H‖aö c‖æqumtfak ( ) 0= sfk

ώ 4ώ

<

,

4 ?

>

b x < mhlet mu

E + x < x

mhlet mu lafks uke sacutfak sur c‖fktervmcce

Oarrfiæ he c‖exerofoe 2

_aft j cm jakotfak hæjfkfe sur c‖fktervmcce F

+

< R pmr

pmr j ( x )

<

x

x

∔

+

=

=

=) Ak m 0

j est oaktfkue sur F

j est hærfvmdce sur F

+

< R ( omr j est uke jakotfak balairmpbfque )

+

et ak m ( ∎ x R ) j ‴( x )

+

< R et ak m

Hako j est strfotelekt orafssmkte sur F

^mr sufte j mhlet uke jakotfak ræofpraque

.

∂ < ;

+

< R

4

( x + = ) 4

hæjfkfe sur c‖ fktervmcce @ j ( F )

j ∔= hæjfkfe sur c‖ fktervmcce

>

< vers F

---

+

Xec que 0 @ < j ( R ) < j ( W >, +∛ W) < j ( > ), cfl j ( x ) < W ∔=,=

W

4) Ak m 0

∔=

y < j ( x ) x < j ( y )

⇘ +

x ∂ @ < W ∔ =,=

W y ∂ F < R

x

⇘ <

y

y

xy x y

y xy x

( )

∔

+

=

=

⇘ + < ∔

⇘ ∔ < +

=

=

y = x x =

⇘ ∔ < +

y

⇘ <

Hako 0 ( ∎ x ∂ W =,=

W) j ( x )

x

=

∔

+

=

x

x

∔ = + =

∔ <

=

∔

x

x

↔+∛

4 âle _ofekoes expærflektmces

_ærfe 0 Cflftes et oaktfkuftæ

_aft j cm jakotfak hæjfkfe sur c‖fktervmcce

Oarrfiæ he c‖exerofoe 6

cm jakotfak hæjfkfe sur c‖fktervmcce F < \ =, +∛W

+∛ pmr

pmr ( )

=) Laktraks que cm jakotfak j mhlet uke jakotfak ræofpraque

Ak m 0

4) Ak m 0

est oaktfkue sur F < \ =, +∛

W

j est oaktfkue sur

j est

strfotelekt orafssmkte sur

( x \ W) j ‴( x ) ( x )

sur F < \ =, +∛W

=, 4 = >

∎ ∂ +∛ < ∔ ; )

^mr sufte j mhlet uke jakotfak ræofpraque

vers F .

Xec que 0 @ < j (\ =, +∛ W) < \ ∔ =, +∛ W

∔=

y < j ( x ) x < j ( y )

⇘

x ∂ \ ∔ =, +∛W

y ∂ \ =, +∛

W

4

x y y

⇘ < ∔

4

4

y y x

4 >

⇘ ∔ ∔ <

+∛ ( omr

4

j x < x ∔ 4x

j ∔=

hæjfkfe sur c‖ fktervmcce @ < j ( F )

j ∔= hæjfkfe sur c‖ fktervmcce

---

4 âle _ofekoes expærflektmces

_ærfe 0 Cflftes et oaktfkuftæ

( )

8 + 8x < 8 = + x ; > omr

x ; ∔ =

∊ < + <

Hako

(

) (

4 ∔ 8 = + x

4 + 8 = + x

y < < = ∔ x + = 3 = au y < < = + = + x ; =

4 4

Hako y = =

< + +

∔=

H‖aö ( \ W) ( )

x

x =, j x = + = + x

∎ ∂ ∔ +∛ < + +

)

Oarrfiæ he c‖exerofoe =>

(

M

D

O

4 Ù 2 4 Ù 4 4 Ù

4 4

5 5

5

=42

? => 4 => =5 => =9

4 4 4

=> 9 => 9 => 9

4 4

=> =>

< < < < < <

?

4

? ?

? ∔ ? + =

? ? ? ?

= 6 ∔ ? + = 6 ∔ ? + =

) 4

?

? + = ? ? ?

?

8

( ? =

) ( ?) ? =

+ ∔ + ( ?)

+ =

< < < <

?

( )

8 2 4

?

8

4

4

=

= =

=

4

? 4

4

= = 4 4 ? =

4 ? = =

? 4 8 ? 4 4

9 =

+ + ∔ 8 .2 .4 4 .4 .4

4+

? 4 4 ? ? ? ?

4 4 < 4 < 8

= = 4

= 4 : :

8 .2 . 4

< < < < < <

8

?

8 4

4 8 4

(

H

? ? ? ?

8 +

)

8 Ù ? +

( ?)

? ?

? ?

( ) ∔( )

4 4

? ?

= =: + =4 +

? 6

< < < <

8 ∔ ? 8 ?

? ?

8 ∔ ?

=: + =4 + 6

? ? ?

Oarreotfak he c‖exerofoe ==

=)

x

x

( x )( )

4 + 4 ∔ x x 4

+ 4 + x

4

cfl + 4 ∔ < cfl < cfl

x ↔+∛ x ↔+∛ 4

x ↔+∛

x

+ 4 + x

x

4

+ 4 ∔ x

4

4

< cfl < >

4 x

4

x + 4 + x ↔+∛

x + 4 + x

( omr

4

x cfl ↔+∛ x + 4 + x < +∛ )

---

4 âle _ofekoes expærflektmces

_ærfe 0 Cflftes et oaktfkuftæ

4)

?

( x )

( ) ( )

?

?

?

x ∔ 4 ∔ 4

x ∔ 2 = =

cfl < cfl < cfl < cfl

<

x ↔ 2 x ∔ 2 x ↔ 2 x ↔ 2 x ↔2

x 2 x 4 x 4 x x x x x

=4

∔ + +

4 4 4

? ? 4

? ? ? ?

( ) ( ∔ 2)( + 4 + 8) ( + 4 + 8)

?)

8)

5)

cfl

4 x + : ∔ x + ? 4 x + : ∔ 4 x

+ ? ∔ 4

< cfl

∔

x ∔ = x ∔ = x ∔=

4 x + : ∔ 2 x + ? ∔ 8

< cfl

∔

? ?

x ↔ = x ↔=

Omr

4

( x ∔ = )( 4 x + : + 4 4 x + : + 8) ( x ∔ = ) ( x + ? + 4)

x ↔= ? ?

4 =

< cfl

∔

4

x ↔= ? ?

4 x + : + 4 4 x + : + 8 x + ? + 4

4 =

< ∔

=4 8

∔=

<

=4

4 4

cfl x + ? ∔ 4x + 8 < cfl x = + ∔ 4x

+ 8

x

x

4

↔+∛ ↔+∛

cfl x < +∛

x

↔+∛

cfl = ?

4

8

+ ∔ + < ∔=

x

4 4

↔+∛

x

x

( x + = )

x + =

cfl < cfl < >

x ∔ 4 ∔ 4

:

? 4

4

x ↔+∛ x ↔+∛ 4

?

( x )

?

x

?

< cfl x = + ∔ 4x

+ 8

4

x

↔+∛

x

? 8

< cfl x

= + ∔ 4 +

4 4

x

↔+∛

x

x

omr

< ∔∛

( x + )

4

( x ∔ 4)

?

?

= x =

cfl < cfl < cfl < >

x x

4

x x

8

↔+∛ ↔+∛ x ↔+∛

:)

( x ∔ = )( x + = )

( )( )

∔ = ∔ = + =

< < < < +∛

∔ = ∔ = ∔=

8 4 4

x x x

cfl cfl 8 cfl 8

cfl 8

4

x ↔ = = = =

= =

x ↔ x ↔ x

x x = ( = ) x

x x

↔

x

x

= x

x

;

∔

; ∔

; ; =

Omr 0

cfl x + = < 4

x ↔=

x ; =

+

cfl x ∔ = < >

x ↔=

x ; =

---

4 âle _ofekoes expærflektmces

_ærfe 0 Cflftes et oaktfkuftæ

Oarrfiæ he c‖exerofoe =4

Ak m 0

x

?

x

=4 >

+ ∔ <

Ak pase

x

:

< t

, c‖æqumtfak hevfekt

t

t

: ? :

=4 >

+ ∔ < ⇘

t

t

? 4

=4 >

+ ∔ <

( t )( t 4

t )

4 ? : >

∔ + + <

t au t t

( < ∔ 3 )

4

4 > ? : > =5 >

∔ < + + < ∊

Hako

:

x < 4 < :8

. H‖aö

. H‖aö _ < { :8}

⇘

⇘

⇘ t <

4