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TC = 2x² + 2xy + 2y² + 40

x + y = 8

2x² + 2xy + 2y² + 40 – λ ( x + y – 8 )

TC_x = 0 → 4x + 2y – λ = 0 → 4x + 2y = λ

TC_y = 0 → 2x + 4y – λ = 0 → 2x + 4y = λ

TC_λ = 0 → -x – y + 8 = 0 → x + y = 8

4x + 2y = λ

2x + 4y = λ (-)

2x – 2y = 0

2x = 2y

x + y = 8

x = 4 , y = 4

TC_xx = 4 > 0 , TC_yy = 4 > 0 , TC_xy = 2

∆ = 4 (4) – 2 = 14 > 0

Maka TC minimum pada ( 4 , 4 )

π = 40x – 2x² + xy – 3y² + 20y – 28

x + y = 16

40x – 2x² + xy – 3y² + 20y – 28 – λ (x + y – 16)

π_x = 0 → 40 – 4x + y – λ = 0 → 40 – 4x + y  = λ

π_y = 0 → x – 6y + 20 – λ = 0 → x – 6y + 20 = λ

π_λ = 0 → -x – y – 16 = 0 → x + y = 16

40 – 4x + y  = λ

x – 6y + 20 = λ  (-)

20 – 5x + 7y = 0

-5x + 7y = -20 │x1│-5x + 7y = -20

x + y = 16       │x5│  5x + 5y = 80  (+)

  12y = 60

      y = 5

x + y = 16

x + 5 = 16

x = 11

π_xx = -4 < 0 , π_yy = -6 < 0 , π_xy = 1

∆ = -4 (-6) – 1 = 23 > 0

Maka π maksimum pada ( 11 , 5 )

Π = 40(11) – 2(11)² + (11)(5) – 3(5)² + 20(5) – 28 = 250

TC = ( 5x – 25 )x – ( 2x – 2y + 15 )y + 300

3x + y = 18

( 5x – 25 )x – ( 2x – 2y + 15 )y + 300 – λ ( 3x + y – 18 )

TC_x = 0 → 10x – 25 – 2y – 3λ = 0 → 10x – 25 – 2y = 3λ

TC_y = 0 → -2x + 4y + 15 – λ = 0 → -2x + 4y + 15 = λ

TC_λ = 0 → -3x – y + 18 = 0 → 3x + y = 18

10x – 25 – 2y = 3λ│x1│10x – 2y – 25 = 3λ

-2x + 4y – 15 = λ  │x3│-6x + 12y – 45 = 3λ  (-)

16x – 14y + 20 = 0

16x – 14y = -20

16x – 14y = -20 │x1│ 16x – 14y = -20

  3x + y = 18   │x14│  42x + 14y = 252  (+)

58x = 232

    x = 4

3x + y = 18  

3 (4) + y = 18

12 + y = 18

        y = 6

TC_xx = 10 > 0 , TC_yy = 4 > 0 , TC_xy = -2

∆ = 10 (4) – (-2) = 42

Maka TC minimum pada ( 4 , 6 )

TR = -3x² - 4y² + 6xy + 50x + 26y + 500

x + y = 80

-3x² - 4y² + 6xy + 50x + 26y + 500 – λ ( x + y – 80 )

TR_x = 0 → -6x + 6y + 50 – λ = 0 → -6x + 6y + 50 = λ

TR_y = 0 → -8y + 6x + 26 – λ = 0 → -8y + 6x + 26 = λ

TR_λ = 0 → -x – y + 80 = 0 → x + y = 80

-6x + 6y + 50 = λ

-8y + 6x + 26 = λ  (-)

-12x + 14y + 24 = 0

-12x + 14y = 24│x1│  -12x + 14y = 24

x + y = 80        │x14│  14x + 14y = 1120  (-)

-26x = 1144

     x = 44

x + y = 80

44 + y = 80

        y = 36

-6x + 6y + 50 = λ

-6(44) + 6(36) + 50 = λ

λ = 2

TR_xx = -6 < 0 , TR_yy = -8 < 0 , TR_xy = 6

∆ = -6 (-8) – 6 = 42 > 0

Maka TR maksimum pada ( 44 , 36 )

TR = -3(44)² - 4(36)² + 6(44)(36) + 50(44) + 26(36) + 500 = 2148