How do you optimize #f(x,y)=2x^2+3y^2-4x-5# subject to #x^2+y^2=81#?

To optimize the function (f(x, y) = 2x^2 + 3y^2 - 4x - 5) subject to the constraint (x^2 + y^2 = 81), we can use the method of Lagrange multipliers.

1. Define the Lagrangian function:

[L(x, y, \lambda) = f(x, y) - \lambda(g(x, y) - 81)]

where (g(x, y) = x^2 + y^2).

2. Calculate the partial derivatives of the Lagrangian with respect to (x), (y), and (\lambda) and set them to zero:

[\frac{\partial L}{\partial x} = 4x - 4 - 2\lambda x = 0]

[\frac{\partial L}{\partial y} = 6y - 3 - 2\lambda y = 0]

[\frac{\partial L}{\partial \lambda} = x^2 + y^2 - 81 = 0]

3. Solve the system of equations to find the critical points.

4. Check the endpoints of the region (in this case, the circle (x^2 + y^2 = 81)) to see if they yield a maximum or minimum.

5. Evaluate the function at the critical points and the endpoints to determine the maximum and minimum values of (f(x, y)) subject to the constraint.