Calculate the maxima and minima of the function f(x, y) = x^2 + y^2 − x + y + 2 in the region R = [(x, y) in R^2|x^2 + y^2 <= 8]. First using the hessian method to determine the local maxima and minima and then the lagrange multipliers method to determine the absolute maxima and minima. Prove also that the ones found are the only ones possible. In each step explain the theorems used and why its possible to use them.