Answer:
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
