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------------------------------------------------ Find the minimum value of [tex]f(x,y) = 76x^2 + 29y^2[/tex] subject to the constraint [tex]x^2 + y^2 = 225[/tex].

To find the minimum value of the function f(x, y) = 76x² + 29y² with the constraint x² + y² = 225, we can use the method of Lagrange multipliers. First, we set up the Lagrangian function:

L(x, y, λ) = 76x² + 29y² + λ(x² + y² - 225)

Taking the partial derivatives of L with respect to x, y, and λ and setting them equal to zero, we can solve for the values of x, y, and λ:

Partial derivative with respect to x: 152x + 2λx = 0
Partial derivative with respect to y: 58y + 2λy = 0
Partial derivative with respect to λ: x² + y² - 225 = 0

Solving this system of equations will give us the values of x, y, and λ. Substituting these values back into the original function, we can find the minimum value.

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