Is this right or wrong (The question is the screenshot)
(A) We want to minimize the cost function C(x, y) = 9x + 12y subject to the constraint U(x, y) =x (y + 3) = 128.9 and non-negativity constraints × ≥ 0 and y ≥ 0
We set up the Lagrangian:
L(x, y, λ) = C(x, y) - λ(U(x, y) - 128.9) Now, we calculate the partial derivatives with respect to x, y, and 1:
∂L/∂x = 9 - λ(y + 3)
∂L/∂y = 12 - λX
∂L/λ = 128.9 - U(x, y)
Setting these partial derivatives equal to zero, we have:
- 9 - λ(y + 3) = 0
2 12 - λx = 0
- 128.9 - x(y + 3) = 0
From equation 2, we get λ= 12/x. Substituting A into equations 1 and 3:
- 9 - (12/x)(y + 3) = 0
2. 128.9 - xY - 3x = 0
Now, we can solve this system of equations simultaneously. Let's start by solving equation 2 for X:
128.9 - xy - 3x = 0
xy = 128.9 - 3x
Now, substitute this into equation 1:
9 - (12/x)(y + 3) = 0
9-(12/x)(128.9 - 3x + 3) = 0
9- (12/x)(131.9 -3x) = 0
9 -1,582.8/x + 36x = 0
Multiply through by x to get rid of the fraction:
9x - 1.582.8 ÷ 36x^2 = 0
Combine like terms:
45x^2 - 1.582.8 = 0
Now, solve for x:
45x^2 = 1.582.8
x^2 = 1,582.8/45
x^2 ≈ 35.1733
X ≈√ 35.1733
X= 5.94 (approx]
Now. substitute this value of x back into equation 2 to find y:
12 - λx = 0
12 - (12/x)(5.94) = 0
12 - 71.64/x = 0
71.64/x = 12
x = 71.64/12
x= 5.97 (approx)
Now that we have x and y, we can calculate λ using equation 2:
λ= 12/X
λ= 12/5 97
λ= 2.01 (approx)
(B) show that the solution found does actually minimize the cost function:
To show that this solution minimizes the cost function, we need to compare the cost at this point to the cost at other feasible points. We'll calculate C(x, y) for the found values of x and y:
C(x, y) = 9x + 12y
C(5.94, 5.97) ≈ 9(5.94) + 12(5.97) ≈ 53.46 + 71.64 ≈ S125.10
Now, we need to check that no other feasible point (x, y) results in a lower cost C(x, y) while satisfying the utility constraint. This step may involve mathematical proofs to establish that the found point is indeed the minimum.
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