Answer :
Final answer:
To find the extremum of f(x, y) subject to the constraint x + 8y = 65, we can use the method of Lagrange multipliers. The extremum value is 28, located at (x, y) = (4, 5), and it is a minimum.
Explanation:
To find the extremum of f(x, y) subject to the constraint x + 8y = 65, we can use the method of Lagrange multipliers. First, we form the Lagrangian function L(x, y, λ) = f(x, y) - λ(x + 8y - 65). Then we take the partial derivatives with respect to x, y, and λ and set them equal to zero. Solving the resulting system of equations will give us the extremum point and whether it is a maximum or a minimum.
To solve the system of equations, we have:
∂L/∂x = -2x - λ = 0
∂L/∂y = -2y - 8λ = 0
∂L/∂λ = x + 8y - 65 = 0
Solving these equations, we find x = 4, y = 5, and λ = -1/2. Substituting these values back into f(x, y), we get f(4, 5) = 69 - 4² - 5² = 69 - 16 - 25 = 28. Therefore, the extremum value is 28, located at (x, y) = (4, 5), and it is a minimum.
