To find the absolute maximum and absolute minimum of the function $f(x, y) = x^2 - xy + y^2$ subject to the constraint $x^2 + y^2 = 4$ using Lagrange multipliers, we follow these steps:

Step 1: Define the Lagrange function

The constraint is $g(x, y) = x^2 + y^2 - 4 = 0$.

The Lagrange function is: $\mathcal{L}(x, y, \lambda) = f(x, y) - \lambda (g(x, y)) = (x^2 - xy + y^2) - \lambda (x^2 + y^2 - 4)$

Step 2: Compute the partial derivatives

To find the critical points, we take the partial derivatives of $\mathcal{L}(x, y, \lambda)$ with respect to $x$, $y$, and $\lambda$, and set them equal to zero.

1. $\frac{\partial \mathcal{L}}{\partial x} = 2x - y - \lambda(2x) = 0$
2. (