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Answer :
Final answer:
To find the maximum value of f(x, y) = 85 - (x^2 - y^2) on the line x + 9y = 82, use the method of Lagrange multipliers.
Explanation:
To find the maximum value of f(x, y) = 85 - (x^2 - y^2) on the line x + 9y = 82, we can use the method of Lagrange multipliers. First, we need to rewrite the equation of the line in terms of y: y = (82 - x)/9. Substituting this into f(x, y), we get a new function g(x) = 85 - (x^2 - ((82 - x)/9)^2). Next, we can find the derivative of g(x) with respect to x and set it equal to 0 to find the critical points. By finding the second derivative and evaluating it at the critical points, we can determine if each point is a maximum or minimum. Finally, we can substitute the x-values of the maximum points back into the equation of the line to find the corresponding y-values and calculate the maximum value of f(x, y).
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