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.

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