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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

Extremums

Test de la dérivée seconde Hessien

(3)

Aujourd’hui, nous allons voir

Extremums sur une région bornée Multiplicateur de Lagrange

(4)

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

petite valeur.

(5)

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>

Nous regarderons donc des régions fermées et bornées.

(6)

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

(7)

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

y<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">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</latexit> = 2

(8)

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=">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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=">AAAC/HicjVLJSsRAEH0T93Eb9eglOAiCMCQq6EUQvXhUcJyBcSGJrTZmkpB0BuKg/+BVr97Eq//iH+jNT7C6bMUF0Q7pvH5Vr1JVXX4Sykw5zmPJ6unt6x8YHCoPj4yOjVcmJnezOE8DUQ/iME6bvpeJUEairqQKRTNJhdf2Q9Hwzza0vdERaSbjaEcVidhveyeRPJaBp4hqnK+688XBwmGl6tQcXvZP4BpQhVlbceUFezhCjAA52hCIoAiH8JDR04ILBwlx++gSlxKSbBe4QJm0OXkJ8vCIPaP9hE4tw0Z01jEzVgf0l5DelJQ2ZkkTk19KWP/NZnvOkTX7W+wux9S5FfT1Taw2sQqnxP6le/f8r07XonCMFa5BUk0JM7q6wETJuSs6c/tTVYoiJMRpfET2lHDAyvc+26zJuHbdW4/tT+ypWX0OjG+OZ5OlQIejFh/Zd/kOJdkT7mVBSNHOt0Qj4X4fgJ9gd6HmLtac7aXq2roZjkFMYwZzNAHLWMMmtlDnvlzhGjfWpXVr3Vn3b65WyWim8GVZD68IFp0e</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

z<latexit sha1_base64="UQvW3Kqj/JYDkbpUifzN4rzNUE4=">AAAC/HicjVJNS8NAEH2N3/Wr6tFLsAiCUFIV9CKIXjwq2FbQKkm6rYtpEpKNWIv+B6969SZe/S/+A735E5wdt+IHohuyeftm3mRmdrw4kKlynKec1dc/MDg0PJIfHRufmCxMTVfTKEt8UfGjIEr2PTcVgQxFRUkViP04EW7bC0TNO93S9tqZSFIZhXuqE4t6222Fsil9VxFVu1g/P1paXDkuFJ2Sw8v+CcoGFGHWTlR4xSEaiOAjQxsCIRThAC5Seg5QhoOYuDq6xCWEJNsFLpEnbUZegjxcYk9pb9HpwLAhnXXMlNU+/SWgNyGljXnSROSXENZ/s9mecWTN/ha7yzF1bh36eiZWm1iFE2L/0vU8/6vTtSg0scY1SKopZkZX55soGXdFZ25/qkpRhJg4jRtkTwj7rOz12WZNyrXr3rpsf2ZPzeqzb3wzvJgsBc44aucj+y7foSR7zL3sEFK08y3RSJS/D8BPUF0qlZdLzu5KcWPTDMcwZjGHBZqAVWxgGzuocF+ucYNb68q6s+6th3dXK2c0M/iyrMc3Di6dIA==</latexit> = x2 + 4 dz

dy = 2y

<latexit sha1_base64="cponBKfUBg8L2Lb6t/MldXNqz/c=">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</latexit>

dz

dx = 2x

<latexit sha1_base64="BtcFr3msyYV+Ycqil/RuvQlLjuo=">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</latexit>

dz

dy = 2y

<latexit sha1_base64="cponBKfUBg8L2Lb6t/MldXNqz/c=">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</latexit>

dz

dx = 2x

<latexit sha1_base64="BtcFr3msyYV+Ycqil/RuvQlLjuo=">AAADB3icjVLLbtRAEKyYACG8DDlysbJC4rTyhkhwQYrgwjGR2CRSEkX27GwYxWtb4/Eqi7UfkH/IFa7cUK75DP4AbnwCNc0EARGCsTyuqe5qd/d0XhemcWn6eSG6tnj9xs2lW8u379y9dz9+8HC7qVqr9FBVRWV386zRhSn10BlX6N3a6mySF3onP37l7TtTbRtTlW/crNYHk+yoNGOjMkfqMI73xzZT3ejdvBudzF+snRzGvbSfykqugkEAPYS1WcXfsI8RKii0mECjhCMukKHhs4cBUtTkDtCRs0RG7BpzLFPb0kvTIyN7zP2Ip73Aljz7mI2oFf9S8LVUJnhMTUU/S+z/loi9lcie/VvsTmL63Gb85iHWhKzDW7L/0l16/q/O1+IwxnOpwbCmWhhfnQpRWumKzzz5pSrHCDU5j0e0W2Ilyss+J6JppHbf20zsX8TTs/6sgm+LryFLjalEnf3MvpM7NLTX0ssZkeMut8SRGPw5AFfB9lp/8LSfbq33Nl6G4VjCI6ziCSfgGTbwGpsYMpspzvAeH6LT6GP0KTr/4RotBM0KflvRxXeDEqIW</latexit>

toutes des fonctions croissantes sur leurs intervalles sans point

critique

(9)

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=">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=">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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

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

z<latexit sha1_base64="UQvW3Kqj/JYDkbpUifzN4rzNUE4=">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</latexit> = x2 + 4 min:

min:

y = 1

<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">AAAC+HicjVKxTtxAEH0YCBdCwhHKNBanSFQnO0GCBulEmpREycFJFxTZZjlW+GzLXiOZE59Am7TpEC1/kz8gHZ/A28mCCAgla3n99s288czsxEWqKxMEv6a86ZnZZ3Ot5/MvFl6+Wmwvvd6p8rpMVD/J07wcxFGlUp2pvtEmVYOiVNE4TtVufPTB2nePVVnpPPtimkLtjaNRpg90EhlSn5vN8Fu7E3QDWf5jEDrQgVvbefsaX7GPHAlqjKGQwRCniFDxGSJEgILcHibkSiItdoVTzFNb00vRIyJ7xH3E09CxGc82ZiXqhH9J+ZZU+nhLTU6/ktj+zRd7LZEt+1TsicS0uTX8xi7WmKzBIdl/6W49/1dnazE4wIbUoFlTIYytLnFRaumKzdy/V5VhhIKcxfu0l8SJKG/77IumktptbyOxX4mnZe05cb41frssFY4lanOX/UTuUNNeSC8bIsNdbokjET4cgMdg5103fN8NPq11eltuOFp4gxWscgLW0cNHbKPPbEY4w3f88E68n965d/HH1ZtymmX8tbzLG1BJm8E=</latexit>

y<latexit sha1_base64="XciUfyG+AUC5HcA82jBL1UaWyX4=">AAAC+HicjVKxTtxAEH0YCBdCwhHKNBanSFQnO0GCBulEmpREycFJFxTZZjlW+GzLXiOZE59Am7TpEC1/kz8gHZ/A28mCCAgla3n99s288czsxEWqKxMEv6a86ZnZZ3Ot5/MvFl6+Wmwvvd6p8rpMVD/J07wcxFGlUp2pvtEmVYOiVNE4TtVufPTB2nePVVnpPPtimkLtjaNRpg90EhlSn5vN8Fu7E3QDWf5jEDrQgVvbefsaX7GPHAlqjKGQwRCniFDxGSJEgILcHibkSiItdoVTzFNb00vRIyJ7xH3E09CxGc82ZiXqhH9J+ZZU+nhLTU6/ktj+zRd7LZEt+1TsicS0uTX8xi7WmKzBIdl/6W49/1dnazE4wIbUoFlTIYytLnFRaumKzdy/V5VhhIKcxfu0l8SJKG/77IumktptbyOxX4mnZe05cb41frssFY4lanOX/UTuUNNeSC8bIsNdbokjET4cgMdg5103fN8NPq11eltuOFp4gxWscgLW0cNHbKPPbEY4w3f88E68n965d/HH1ZtymmX8tbzLG1BJm8E=</latexit> = 1 max:

max:

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

y<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">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</latexit> = 2

min:

min:

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

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

max:

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

x<latexit sha1_base64="/gbhdTRUYH6WDhLh6D3AwbwxJQ0=">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</latexit> = 3 z<latexit sha1_base64="8pmKcJTacZexY0ai7MYvQZrgxYg=">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</latexit> = 2

z<latexit sha1_base64="uOSu1Mf7Q1M2PEVXjAoZ8kyoefs=">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</latexit> = 5

z<latexit sha1_base64="Q09HN+rnO8Vli3WYN45sUqCmTTM=">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</latexit> = 10 z<latexit sha1_base64="pq2+kqGk0VdLyW6vPZJ4olN2Zxw=">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</latexit> = 13

z<latexit sha1_base64="8pmKcJTacZexY0ai7MYvQZrgxYg=">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</latexit> = 2 z<latexit sha1_base64="Q09HN+rnO8Vli3WYN45sUqCmTTM=">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</latexit> = 10

z<latexit sha1_base64="uOSu1Mf7Q1M2PEVXjAoZ8kyoefs=">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</latexit> = 5 z<latexit sha1_base64="pq2+kqGk0VdLyW6vPZJ4olN2Zxw=">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</latexit> = 13

(10)

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>

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

z<latexit sha1_base64="UQvW3Kqj/JYDkbpUifzN4rzNUE4=">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</latexit> = x2 + 4 min:

min:

y = 1

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

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

max:

y<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">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</latexit> = 2

y<latexit sha1_base64="kQcg/ch0xdVy1wewEHAYoOcuTpU=">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</latexit> = 2

min:

min:

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="2HGZBjIElNiaStoRwwtiS3u16tE=">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</latexit> = 1 max:

max:

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

x<latexit sha1_base64="/gbhdTRUYH6WDhLh6D3AwbwxJQ0=">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</latexit> = 3 z<latexit sha1_base64="8pmKcJTacZexY0ai7MYvQZrgxYg=">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</latexit> = 2

z<latexit sha1_base64="uOSu1Mf7Q1M2PEVXjAoZ8kyoefs=">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</latexit> = 5

z<latexit sha1_base64="Q09HN+rnO8Vli3WYN45sUqCmTTM=">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</latexit> = 10 z<latexit sha1_base64="pq2+kqGk0VdLyW6vPZJ4olN2Zxw=">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</latexit> = 13

z<latexit sha1_base64="8pmKcJTacZexY0ai7MYvQZrgxYg=">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</latexit> = 2 z<latexit sha1_base64="Q09HN+rnO8Vli3WYN45sUqCmTTM=">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</latexit> = 10

z<latexit sha1_base64="uOSu1Mf7Q1M2PEVXjAoZ8kyoefs=">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</latexit> = 5 z<latexit sha1_base64="pq2+kqGk0VdLyW6vPZJ4olN2Zxw=">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</latexit> = 13

(11)

Exemple

Trouver les extrémums absolus de la fonction f (x, y) = x2 2y2

<latexit sha1_base64="hjJ60nDgpe1VWUsukkAVcIEpKW0=">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</latexit> sur le triangle passant par r<latexit sha1_base64="DzPqe5Ocx0hJUZWW7pWbI6d3sfY=">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</latexit> f = (2x, 4y) (0,<latexit sha1_base64="ylWp8btNhNC9mIsUybv58M8g8pw=">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</latexit> 0) 2/ T

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

La droite passant par et (1,<latexit sha1_base64="Qr5mlQfbDPdu98J4Aky6p3yOQHQ=">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</latexit> 2) (2,<latexit sha1_base64="2SHiISk7zfGT/P1dHPh2Xs2ix70=">AAAC+nicjVJNS+RAEH1md/1cdVyPXoKDoCBDooIexb14VNhRQWVJYquNmSQkHWEY/Q1e9bo38eqf8R/ozZ+wr8tWdEXcDum8flWvUlVdcZHqygTBXZ/35eu3/oHBoeGR76Nj442JH1tVXpeJaid5mpc7cVSpVGeqbbRJ1U5RqqgTp2o7Pvlp7dunqqx0nv0y3ULtd6KjTB/qJDKk2rML8+Hc70YzaAWy/PcgdKAJtzbyxiP2cIAcCWp0oJDBEKeIUPHZRYgABbl99MiVRFrsCucYpraml6JHRPaE+xFPu47NeLYxK1En/EvKt6TSxww1Of1KYvs3X+y1RLbsR7F7EtPm1uU3drE6ZA2OyX6me/b8X52txeAQK1KDZk2FMLa6xEWppSs2c/9VVYYRCnIWH9BeEieifO6zL5pKare9jcR+L56WtefE+dZ4cFkqnErU7kv2PblDTXshvewSGe5ySxyJ8N8BeA+2FlrhYivYXGqurrnhGMQUpjHLCVjGKtaxgTaz0bjAJa68M++Pd+3dPLl6fU4ziTfLu/0Lismbzg==</latexit> 1)

y<latexit sha1_base64="hHAj28YCA2uBRRmZi8PlfkV6B14=">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</latexit> = x + b 2 =<latexit sha1_base64="ut/AsmBbflVzXN0IZ7WU2NqUJmY=">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</latexit> 1 + b b<latexit sha1_base64="jwBa0McVvgKdyhwJyn2nWFxQqxY=">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</latexit> = 3 y<latexit sha1_base64="+pUELiRsRAaCjxdcXWgllLWbwv8=">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</latexit> = x + 3

(1, 1)

<latexit sha1_base64="7e0TCshlfZHCYIh801OHBpKdhbQ=">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</latexit>

(1,<latexit sha1_base64="Qr5mlQfbDPdu98J4Aky6p3yOQHQ=">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</latexit> 2)

(2, 1)

<latexit sha1_base64="2SHiISk7zfGT/P1dHPh2Xs2ix70=">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</latexit>

(12)

Exemple

Trouver les extrémums absolus de la fonction f (x, y) = x2 2y2

<latexit sha1_base64="hjJ60nDgpe1VWUsukkAVcIEpKW0=">AAADB3icjVJNT9RQFD0UVD5EKyzZNE5MMNFJZzSBjQmBjUtIGCBBIG15Ay/TaZv2dUIz8Qf4H9zKlh1x68/wH+iOn8B51wdRicHX9PW8c++5vfe+GxeprkwYfp/wJqcePHw0PTM793j+yVP/2cJOlddlonpJnublXhxVKtWZ6hltUrVXlCoaxqnajQcb1r47UmWl82zbNIU6GEYnme7rJDKkjny/v3z2Kmhevjs77L7uNofdI78VtkNZwV3QcaAFtzZz/wofcIwcCWoMoZDBEKeIUPHZRwchCnIHGJMribTYFT5iltqaXooeEdkB9xOe9h2b8WxjVqJO+JeUb0llgBfU5PQrie3fArHXEtmy/4o9lpg2t4bf2MUakjU4JXuf7sbzf3W2FoM+VqUGzZoKYWx1iYtSS1ds5sFvVRlGKMhZfEx7SZyI8qbPgWgqqd32NhL7D/G0rD0nzrfGT5elwkiiNrfZj+UONe2F9LIhMtzlljgSnb8H4C7Y6bY7b9rh1tvW2robjmks4TmWOQErWMN7bKLHbEb4jC849z55F96l9/WXqzfhNIv4Y3nfrgGNV6Au</latexit> sur le triangle passant par

(1, 1)

<latexit sha1_base64="7e0TCshlfZHCYIh801OHBpKdhbQ=">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</latexit>

(1,<latexit sha1_base64="Qr5mlQfbDPdu98J4Aky6p3yOQHQ=">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</latexit> 2)

(2, 1)

<latexit sha1_base64="2SHiISk7zfGT/P1dHPh2Xs2ix70=">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</latexit>

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

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

y<latexit sha1_base64="+pUELiRsRAaCjxdcXWgllLWbwv8=">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</latexit> = x + 3

z<latexit sha1_base64="uldzeinsdK2mQ22FPhthCep7kUE=">AAAC/XicjVJNS8NAEH3G7++qRy/BInixJFXQiyB68ahgtaBVknSri2kSko0Qi/gfvOrVm3j1t/gP9OZPcHZcRS2iG7J5+2beZGZ2/CSUmXKcpx6rt69/YHBoeGR0bHxisjQ1vZfFeRqIWhCHcVr3vUyEMhI1JVUo6kkqvLYfin3/bFPb989Fmsk42lVFIhpt7ySSLRl4iqj6xZq7WC2OqselslNxeNndwDWgDLO249IrDtFEjAA52hCIoAiH8JDRcwAXDhLiGugQlxKSbBe4xAhpc/IS5OERe0b7CZ0ODBvRWcfMWB3QX0J6U1LamCdNTH4pYf03m+05R9bsb7E7HFPnVtDXN7HaxCqcEvuX7sPzvzpdi0ILq1yDpJoSZnR1gYmSc1d05vaXqhRFSIjTuEn2lHDAyo8+26zJuHbdW4/tz+ypWX0OjG+OF5OlwDlHLT6z7/AdSrIn3MuCkKKdb4lGwv05AN1gr1pxlyrOznJ5fcMMxxBmMYcFmoAVrGML26jxzV/jBrfWlXVn3VsP765Wj9HM4NuyHt8AtW2dXA==</latexit> = 1 2y2

[1,<latexit sha1_base64="Jk5zfxhnKWdiELcu1N8PxPV+oFo=">AAAC+nicjVJNT9tAEH2YttD0K8CxF6tRJQ5VZIdKcET0wjFIJCClUWU7C13FsS17jWQFfgPX9tob6rV/hn8AN34Cb6dL1YKqdi2v376ZN56ZnbhIdWWC4HLBW3z0+MnS8tPWs+cvXr5qr6wOq7wuEzVI8jQvD+OoUqnO1MBok6rDolTRLE7VQTz9YO0HJ6qsdJ7tm6ZQ41l0nOkjnUSG1GAUvuuNP7U7QTeQ5T8EoQMduNXP2zf4iAlyJKgxg0IGQ5wiQsVnhBABCnJjzMmVRFrsCmdoUVvTS9EjIjvlfszTyLEZzzZmJeqEf0n5llT6eEtNTr+S2P7NF3stkS37t9hziWlza/iNXawZWYPPZP+lu/P8X52txeAIW1KDZk2FMLa6xEWppSs2c/+3qgwjFOQsntBeEieivOuzL5pKare9jcR+JZ6WtefE+da4dlkqnEjU5lf2c7lDTXshvWyIDHe5JY5EeH8AHoJhrxtudIO9953tHTccy3iNN1jnBGxiG7voY8BsNM7xBV+9U++bd+F9/+nqLTjNGv5Y3o9blGCcNQ==</latexit> 2]

[1,<latexit sha1_base64="Jk5zfxhnKWdiELcu1N8PxPV+oFo=">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</latexit> 2]

z<latexit sha1_base64="6KY9YvEjSZpSA6cqfx0+g0DrsIQ=">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</latexit> = x2 2

=<latexit sha1_base64="lrY8bCw6wcz5AucaId7tmbYrhwA=">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</latexit> x2 + 12x 9 [1, 2]

<latexit sha1_base64="Jk5zfxhnKWdiELcu1N8PxPV+oFo=">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</latexit>

=<latexit sha1_base64="ZL1cdb7dppSi4qAiui4pMOJTxBw=">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</latexit> x2 2( x + 3)2 z<latexit sha1_base64="25kPWV9cuwT0Ot/PY9FccEOo+e0=">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</latexit>

dz

dy = 4y

<latexit sha1_base64="KoQ/Zrxt4bY/tS2S5jWV3IcaL8o=">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</latexit>

dz

dx = 2x

<latexit sha1_base64="BtcFr3msyYV+Ycqil/RuvQlLjuo=">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</latexit>

dz

dx = 2x + 12

<latexit sha1_base64="la974cEHCvitWrjdP9Wg+Fvlh3g=">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</latexit>

croissante décroissante

croissante

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