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20 May, 05:22

The function f (x, y) = 2x + 2y has an absolute maximum value and absolute minimum value subject to the constraint 9x^2 - 9xy + 9y^2 = 25. Use Lagrange multipliers to find these values.

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  1. 20 May, 05:48
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    the minimum is located in x = - 5/3, y = - 5/3

    Step-by-step explanation:

    for the function

    f (x, y) = 2x + 2y

    we define the function g (x) = 9x² - 9xy + 9y² - 25 (for g (x) = 0 we get the constrain)

    then using Lagrange multipliers f (x) is maximum when

    fx-λgx (x) = 0 → 2 - λ (9*2x - 9*y) = 0 →

    fy-λgy (x) = 0 → 2 - λ (9*2y - 9*x) = 0

    g (x) = 0 → 9x² - 9xy + 9y² - 25 = 0

    subtracting the second equation to the first we get:

    2 - λ (9*2y - 9*x) - (2 - λ (9*2x - 9*y)) = 0

    - 18*y + 9*x + 18*x - 9*y = 0

    27*y = 27 x → x=y

    thus

    9x² - 9xy + 9y² - 25 = 0

    9x² - 9x² + 9x² - 25 = 0

    9x² = 25

    x = ±5/3

    thus

    y = ±5/3

    for x=5/3 and y=5/3 → f (x) = 20/3 (maximum), while for x = - 5/3, y = - 5/3 → f (x) = - 20/3 (minimum)

    finally evaluating the function in the boundary, we know because of the symmetry of f and g with respect to x and y that the maximum and minimum are located in x=y

    thus the minimum is located in x = - 5/3, y = - 5/3
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