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2.6 MULTIPLICATEURS DE LAGRANGE

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Texte intégral

(1)

cours 14

2.6 MULTIPLICATEURS

DE LAGRANGE

(2)

Au dernier cours, nous avons vu

(3)

Au dernier cours, nous avons vu

Extremums

(4)

Au dernier cours, nous avons vu

Extremums

Test de la dérivée seconde

(5)

Au dernier cours, nous avons vu

Extremums

Test de la dérivée seconde Hessien

(6)

Aujourd’hui, nous allons voir

(7)

Aujourd’hui, nous allons voir

Extremums sur une région bornée

(8)

Aujourd’hui, nous allons voir

Extremums sur une région bornée Multiplicateur de Lagrange

(9)

Avec les fonctions à une variable, il arrivait parfois qu’on veuille trouver le maximum ou le minimum absolu sur un intervalle.

(10)

Avec les fonctions à une variable, il arrivait parfois qu’on veuille trouver le maximum ou le minimum absolu sur un intervalle.

On commence par trouver les extremums relatifs comme

précédemment.

(11)

Avec les fonctions à une variable, il arrivait parfois qu’on veuille trouver le maximum ou le minimum absolu sur un intervalle.

On commence par trouver les extremums relatifs comme

précédemment.

Et on compare avec les

points frontière pour trouver la plus grande et la plus

(12)

Avec les fonctions à deux variables, c’est un peu la même chose, mais on doit préciser quel type de généralisation d’un intervalle on veut

considérer.

(13)

Avec les fonctions à deux variables, c’est un peu la même chose, mais on doit préciser quel type de généralisation d’un intervalle on veut

considérer.

Un ensemble sera dit fermé s’il contient tous ces points frontière.

(14)

Avec les fonctions à deux variables, c’est un peu la même chose, mais on doit préciser quel type de généralisation d’un intervalle on veut

considérer.

Un ensemble sera dit fermé s’il contient tous ces points frontière.

(le concept d’ensemble fermé est légèrement plus compliqué que ça, mais nous nous contenterons de cette définition)

(15)

Avec les fonctions à deux variables, c’est un peu la même chose, mais on doit préciser quel type de généralisation d’un intervalle on veut

considérer.

Un ensemble sera dit fermé s’il contient tous ces points frontière.

(le concept d’ensemble fermé est légèrement plus compliqué que ça, mais nous nous contenterons de cette définition)

Un ensemble est dit borné s’il existe un nombre tel queA<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> r<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(16)

Avec les fonctions à deux variables, c’est un peu la même chose, mais on doit préciser quel type de généralisation d’un intervalle on veut

considérer.

Un ensemble sera dit fermé s’il contient tous ces points frontière.

(le concept d’ensemble fermé est légèrement plus compliqué que ça, mais nous nous contenterons de cette définition)

Un ensemble est dit borné s’il existe un nombre tel queA<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> r<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

A ⇢ B = (x, y) 2 R2|x2 + y2 < r

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(17)

Avec les fonctions à deux variables, c’est un peu la même chose, mais on doit préciser quel type de généralisation d’un intervalle on veut

considérer.

Un ensemble sera dit fermé s’il contient tous ces points frontière.

(le concept d’ensemble fermé est légèrement plus compliqué que ça, mais nous nous contenterons de cette définition)

Un ensemble est dit borné s’il existe un nombre tel queA<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit> r<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

A ⇢ B = (x, y) 2 R2|x2 + y2 < r

<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

(18)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

(19)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle

(20)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle

(21)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle

(22)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle r<latexit sha1_base64="iLlqqU1RP58PPXe9AxoXJ005l8w=">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</latexit> f = (2x, 2y)

(23)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle r<latexit sha1_base64="iLlqqU1RP58PPXe9AxoXJ005l8w=">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</latexit> f = (2x, 2y)

Le seul point critique

(24)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle r<latexit sha1_base64="iLlqqU1RP58PPXe9AxoXJ005l8w=">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</latexit> f = (2x, 2y)

Le seul point critique (0,<latexit sha1_base64="ArioZHqqOoghGJcPi/OG8/U6gW4=">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</latexit> 0) 2/ [1, 3] ⇥ [1, 2]

(25)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle r<latexit sha1_base64="iLlqqU1RP58PPXe9AxoXJ005l8w=">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</latexit> f = (2x, 2y)

Le seul point critique (0,<latexit sha1_base64="ArioZHqqOoghGJcPi/OG8/U6gW4=">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</latexit> 0) 2/ [1, 3] ⇥ [1, 2]

(26)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle r<latexit sha1_base64="iLlqqU1RP58PPXe9AxoXJ005l8w=">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</latexit> f = (2x, 2y)

Le seul point critique (0,<latexit sha1_base64="ArioZHqqOoghGJcPi/OG8/U6gW4=">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</latexit> 0) 2/ [1, 3] ⇥ [1, 2]

On doit aussi regarder sur la frontière

(27)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle

(28)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">AAADC3icjVLLTtxAECwcEl55LIEbFyurSDlEKxsiwRGRC0eQWEBaVsg2Axnhl+wx0rLiE/gHruTKDXHlI/IH4cYnUNMMiARFYSyPa6q72t09HZeprk0Q/BrxXo2+fjM2PjE59fbd+w+t6Y+bddFUieomRVpU23FUq1Tnqmu0SdV2Wakoi1O1FR9+t/atI1XVusg3zKBU/Sw6yPW+TiJDarc12/PDr/5Cf8foTNVymO/vttpBJ5DlPwehA224tVa0brGDPRRI0CCDQg5DnCJCzaeHEAFKcn0MyVVEWuwKJ5iktqGXokdE9pD7AU89x+Y825i1qBP+JeVbUenjMzUF/Spi+zdf7I1Etuy/Yg8lps1twG/sYmVkDX6Q/Z/uwfOlOluLwT6WpAbNmkphbHWJi9JIV2zm/pOqDCOU5Czeo70iTkT50GdfNLXUbnsbif23eFrWnhPn2+DGZalwJFEHj9kP5Q417aX0ckBkuMstcSTCvwfgOdic74QLnWD9W3t5xQ3HOObwCV84AYtYxirW0GU2xzjDOX56p96Fd+ld3bt6I04zgz+Wd30HlnihQg==</latexit> 3] ⇥ [1, 2]

dans le rectangle

x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1

(29)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle

x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1 y<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit> = 1

(30)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle

x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1

x<latexit sha1_base64="/gbhdTRUYH6WDhLh6D3AwbwxJQ0=">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</latexit> = 3

y<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit> = 1

(31)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">AAADB3icjVJNT9RQFD0UVD5EKyzZNE5MMJpJZzSBjQmBjUtIGCBBIG15Ay/TaZv2dUIz8Qf4H9zKlh1x68/wH+iOn8B51wdRicHX9PW8c++5vfe+GxeprkwYfp/wJqcePHw0PTM793j+yVP/2cJOlddlonpJnublXhxVKtWZ6hltUrVXlCoaxqnajQcb1r47UmWl82zbNIU6GEYnme7rJDKkjny/v3z2OmhevgvODruvmsPukd8K26Gs4C7oONCCW5u5f4UPOEaOBDWGUMhgiFNEqPjso4MQBbkDjMmVRFrsCh8xS21NL0WPiOyA+wlP+47NeLYxK1En/EvKt6QywAtqcvqVxPZvgdhriWzZf8UeS0ybW8Nv7GINyRqckr1Pd+P5vzpbi0Efq1KDZk2FMLa6xEWppSs28+C3qgwjFOQsPqa9JE5EedPnQDSV1G57G4n9h3ha1p4T51vjp8tSYSRRm9vsx3KHmvZCetkQGe5ySxyJzt8DcBfsdNudN+1w621rbd0NxzSW8BzLnIAVrOE9NtFjNiN8xhece5+8C+/S+/rL1ZtwmkX8sbxv11j+oBo=</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle

x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1 y<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit> = 1

y = 2

(32)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">AAADB3icjVJNT9RQFD0UVD5EKyzZNE5MMJpJZzSBjQmBjUtIGCBBIG15Ay/TaZv2dUIz8Qf4H9zKlh1x68/wH+iOn8B51wdRicHX9PW8c++5vfe+GxeprkwYfp/wJqcePHw0PTM793j+yVP/2cJOlddlonpJnublXhxVKtWZ6hltUrVXlCoaxqnajQcb1r47UmWl82zbNIU6GEYnme7rJDKkjny/v3z2OmhevgvODruvmsPukd8K26Gs4C7oONCCW5u5f4UPOEaOBDWGUMhgiFNEqPjso4MQBbkDjMmVRFrsCh8xS21NL0WPiOyA+wlP+47NeLYxK1En/EvKt6QywAtqcvqVxPZvgdhriWzZf8UeS0ybW8Nv7GINyRqckr1Pd+P5vzpbi0Efq1KDZk2FMLa6xEWppSs28+C3qgwjFOQsPqa9JE5EedPnQDSV1G57G4n9h3ha1p4T51vjp8tSYSRRm9vsx3KHmvZCetkQGe5ySxyJzt8DcBfsdNudN+1w621rbd0NxzSW8BzLnIAVrOE9NtFjNiN8xhece5+8C+/S+/rL1ZtwmkX8sbxv11j+oBo=</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1

x<latexit sha1_base64="/gbhdTRUYH6WDhLh6D3AwbwxJQ0=">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</latexit> = 3

y = 1

<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit>

y = 2

<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">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</latexit>

(33)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1

x<latexit sha1_base64="/gbhdTRUYH6WDhLh6D3AwbwxJQ0=">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</latexit> = 3

y = 1

<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit>

y = 2

<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">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</latexit>

z<latexit sha1_base64="aKqMe4y5U66vHNuESQwXGiW0lxM=">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</latexit> = 1 + y2

(34)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1

x<latexit sha1_base64="/gbhdTRUYH6WDhLh6D3AwbwxJQ0=">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</latexit> = 3

y = 1

<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit>

y = 2

<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">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</latexit>

z<latexit sha1_base64="aKqMe4y5U66vHNuESQwXGiW0lxM=">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</latexit> = 1 + y2

(35)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1

x<latexit sha1_base64="/gbhdTRUYH6WDhLh6D3AwbwxJQ0=">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</latexit> = 3

y = 1

<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit>

y = 2

<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">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</latexit>

z<latexit sha1_base64="aKqMe4y5U66vHNuESQwXGiW0lxM=">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</latexit> = 1 + y2

z<latexit sha1_base64="f4sHhyYu1a+IrxLefRGWW/ZXVno=">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</latexit> = 9 + y2

(36)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">AAADC3icjVLLTtxAECwcEl55LIEbFyurSDlEKxsiwRGRC0eQWEBaVsg2Axnhl+wx0rLiE/gHruTKDXHlI/IH4cYnUNMMiARFYSyPa6q72t09HZeprk0Q/BrxXo2+fjM2PjE59fbd+w+t6Y+bddFUieomRVpU23FUq1Tnqmu0SdV2Wakoi1O1FR9+t/atI1XVusg3zKBU/Sw6yPW+TiJDarc12/PDr/5Cf8foTNVymO/vttpBJ5DlPwehA224tVa0brGDPRRI0CCDQg5DnCJCzaeHEAFKcn0MyVVEWuwKJ5iktqGXokdE9pD7AU89x+Y825i1qBP+JeVbUenjMzUF/Spi+zdf7I1Etuy/Yg8lps1twG/sYmVkDX6Q/Z/uwfOlOluLwT6WpAbNmkphbHWJi9JIV2zm/pOqDCOU5Czeo70iTkT50GdfNLXUbnsbif23eFrWnhPn2+DGZalwJFEHj9kP5Q417aX0ckBkuMstcSTCvwfgOdic74QLnWD9W3t5xQ3HOObwCV84AYtYxirW0GU2xzjDOX56p96Fd+ld3bt6I04zgz+Wd30HlnihQg==</latexit> 3] ⇥ [1, 2]

dans le rectangle x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1

x<latexit sha1_base64="/gbhdTRUYH6WDhLh6D3AwbwxJQ0=">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</latexit> = 3

y = 1

<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit>

y = 2

<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">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</latexit>

z<latexit sha1_base64="aKqMe4y5U66vHNuESQwXGiW0lxM=">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</latexit> = 1 + y2

z<latexit sha1_base64="f4sHhyYu1a+IrxLefRGWW/ZXVno=">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</latexit> = 9 + y2

(37)

Exemple

Trouver la valeur maximale de f (x, y) = x2 + y2

<latexit sha1_base64="JWZAPgTktgF3wKQjqb9E6TRDG2Y=">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</latexit>

[1,<latexit sha1_base64="Bo/lOC4hBCkk6rObkk1eetFP6Yk=">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</latexit> 3] ⇥ [1, 2]

dans le rectangle x<latexit sha1_base64="2HGZBjIElNiaStoRwwtiS3u16tE=">AAAC+HicjVJNT9RQFD1TUMdRdISlm4aJiatJqya6MSG6YTkE5iMZCGnLY3ih0zbtK3Gc+BPcypYdceu/4R/Ajp/gedcHEYjB1/T1vHPvub33vhsXqa5MEJw1vIXFBw8fNR+3njxdeva8/WJ5UOV1mah+kqd5OYqjSqU6U32jTapGRamiaZyqYXz42dqHR6qsdJ5tmVmhdqbRJNP7OokMqc0vH8PddifoBrL8uyB0oAO3enn7EtvYQ44ENaZQyGCIU0So+IwRIkBBbgdzciWRFrvCN7Soreml6BGRPeQ+4Wns2IxnG7MSdcK/pHxLKn28oianX0ls/+aLvZbIlv1X7LnEtLnN+I1drClZgwOy9+muPP9XZ2sx2McHqUGzpkIYW13iotTSFZu5/1dVhhEKchbv0V4SJ6K86rMvmkpqt72NxH4unpa158T51rhwWSocSdTZdfZzuUNNeyG9nBEZ7nJLHInw9gDcBYM33fBtN9h411n75IajiZdYxWtOwHusYR099JnNBN/xA8feV+/EO/V+/nH1Gk6zghvL+/UbTbWbwA==</latexit> = 1

x<latexit sha1_base64="/gbhdTRUYH6WDhLh6D3AwbwxJQ0=">AAAC+HicjVLBbtNAEH01LU1LgQBHLlYjpJ4ih1Sil0oRvXBMVdJWKhWynW26imNb9jqqifiEXtsrN8S1f8MftLd+Qt8OWwRECNby+u2beeOZ2YnyRJcmCL4veA8Wlx4uN1ZWH609fvK0+ez5fplVRawGcZZkxWEUlirRqRoYbRJ1mBcqnESJOojGO9Z+MFVFqbP0valzdTwJR6k+0XFoSO2dbXc/NltBO5Dlz4OOAy241c+at/iAITLEqDCBQgpDnCBEyecIHQTIyR1jRq4g0mJX+IxVait6KXqEZMfcRzwdOTbl2cYsRR3zLwnfgkofr6jJ6FcQ27/5Yq8ksmX/FnsmMW1uNb+RizUha3BK9l+6e8//1dlaDE6wJTVo1pQLY6uLXZRKumIz93+pyjBCTs7iIe0FcSzK+z77oimldtvbUOzX4mlZe46db4Ubl6XCVKLWP7OfyR1q2nPpZU1kuMstcSQ6fw7APNh/3e5028HuZqv31g1HAy+xjg1OwBv08A59DJjNCOe4wKX3yfviffW+/XD1FpzmBX5b3tUdUtmbwg==</latexit> = 3

y = 1

<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">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</latexit>

y = 2

<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">AAAC+HicjVLLTtxAECwMhGeSDRy5WKyQOK28BIlckFBy4QiCBSRAkW2GzQivbdnjlcyKT8g1uXJDXPkb/gBufAI1zYB4CCVjeVxT3dXu7ukoT3RpguB6yBseGf0wNj4xOTX98dPnxpeZnTKrilh14izJir0oLFWiU9Ux2iRqLy9U2IsStRud/LD23b4qSp2l26bO1WEv7Kb6WMehIbVVry79bDSDViDLfwvaDjTh1kbWuMMBjpAhRoUeFFIY4gQhSj77aCNATu4QA3IFkRa7whkmqa3opegRkj3h3uVp37EpzzZmKeqYf0n4FlT6WKAmo19BbP/mi72SyJZ9L/ZAYtrcan4jF6tH1uAX2X/pHj3/V2drMTjGN6lBs6ZcGFtd7KJU0hWbuf+sKsMIOTmLj2gviGNRPvbZF00ptdvehmK/EU/L2nPsfCvcuiwV+hK1fsp+IHeoac+llzWR4S63xJFovx6At2BnqdX+2go2l5tr391wjGMO81jkBKxgDevYQIfZdPEbf/DXO/XOvQvv8sHVG3KaWbxY3tU9Utubwg==</latexit>

z<latexit sha1_base64="aKqMe4y5U66vHNuESQwXGiW0lxM=">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</latexit> = 1 + y2

z<latexit sha1_base64="f4sHhyYu1a+IrxLefRGWW/ZXVno=">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</latexit> = 9 + y2

z<latexit sha1_base64="7Qa6SOGbzveE1e1ngflOwgGSfSA=">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</latexit> = x2 + 1

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