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Answer :
Final answer:
To find the extreme values of f(x, y) subject to the constraint x + 8y = 65, we can use the method of Lagrange multipliers. We need to set up the Lagrangian function, find the critical points by setting the partial derivatives of the Lagrangian equal to zero, and evaluate f(x, y) at these points to determine if they correspond to a maximum or minimum.
Explanation:
To find the extreme values of f(x, y) subject to the constraint x + 8y = 65, we can use the method of Lagrange multipliers. We need to set up the Lagrangian function L(x, y, λ) = f(x, y) - λ(g(x, y) - c), where g(x, y) is the constraint equation, c is the constant in the constraint equation, and λ is the Lagrange multiplier. In this case, the Lagrangian is L(x, y, λ) = 69 [tex]- x^2 - y^2[/tex] 65).
Next, we need to find the partial derivatives of L with respect to x, y, and λ, and set them equal to zero. This will give us a system of equations to solve for the critical points. Solving this system will give us x and y values, as well as the value of λ.
Finally, we need to evaluate f(x, y) at the critical points to determine if they correspond to a maximum or minimum. If the second partial derivatives of f are positive at a critical point, then it is a minimum. If the second partial derivatives are negative, then it is a maximum.
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