Using Lagrange multipliers find the absolute maximum and absolute minimum of f(x,y) = x2 − xy + y2 subject to the constraint x2 + y2 = 4

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. \(

Using Lagrange multipliers find the absolute maximum and absolute minimum of f(x,y) = x^2 − xy + y^2 subject to the constraint x^2 + y^2 = 4

Learn how to find the absolute maximum and minimum of the function f(x, y) = x^2 − xy + y^2 subject to the constraint x^2 + y^2 = 4 using Lagrange multipliers. This detailed step-by-step solution involves multivariable calculus concepts and the Lagrange multiplier method. Suitable for undergraduate students.

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