To solve this problem using the Lagrange multiplier method, we need to find the minimum value of the function $f(x, y) = x^2 + xy + y^2$ under the constraint $g(x, y) = x^2 - xy - 6y^2 - 1 = 0$.

Step-by-Step Solution:

1. Define the Objective and Constraint Functions:

   • Objective function to minimize: $f(x, y) = x^2 + xy + y^2$
   • Constraint function: $g(x, y) = x^2 - xy - 6y^2 - 1 = 0$

2. Formulate the Lagrange Function: Define the Lagrange function $\mathcal{L}(x, y, \lambda)$ as: $\mathcal{L}(x, y, \lambda) = f(x, y) - \lambda \cdot g(x, y)$ Substituting $f(x, y)$ and $g(x, y)$ into this, we get: $\mathcal{L}(x, y, \lambda) = x^2 + xy + y^2 - \lambda (x^2 - xy - 6y^2 - 1)$

3. Compute Partial Derivatives: To use the method of Lagrange multipliers, set the partial derivatives of $\mathcal{L}$ with respect to $x$, $y$, and $\lambda$ equal to zero.

   • Partial derivative with respect to $x$: $\frac{\partial \mathcal{L}}{\partial x} = 2x + y - \lambda (2x - y) = 0$

   • Partial derivative with respect to $y$: $\frac{\partial \mathcal{L}}{\partial y} = x + 2y - \lambda (-x - 12y) = 0$

   • Partial derivative with respect to $\lambda$: $\frac{\partial \mathcal{L}}{\partial \lambda} = -(x^2 - xy - 6y^2 - 1) = 0$ This yields the constraint equation: $x^2 - xy - 6y^2 = 1$

4. Solve the System of Equations: Now, we have a system of three equations: $2x + y - \lambda (2x - y) = 0$ $x + 2y - \lambda (-x - 12y) = 0$ $x^2 - xy - 6y^2 = 1$

   Solving these equations simultaneously will give values for $x$, $y$, and $\lambda$ that satisfy the constraint while minimizing $f(x, y)$.

5. Substitute Values and Calculate Minimum: Substitute possible values of $x$ and $y$ obtained from the system back into $f(x, y)$ to determine the minimum value.

Derivatives Used in This Solution:

• Derivative of $x^2$ with respect to $x$: $2x$
• Derivative of $xy$ with respect to $x$ and $y$: $y$ and $x$ respectively
• Derivative of $y^2$ with respect to $y$: $2y$

Would you like further details on solving the system of equations, or have any questions?

Follow-Up Questions:

1. What other methods can solve constrained optimization problems besides Lagrange multipliers?
2. How do we confirm if a solution obtained from Lagrange multipliers is a minimum or maximum?
3. Can Lagrange multipliers be used for multi-variable constraints?
4. What are real-life applications of using Lagrange multipliers in optimization?
5. How does the constraint affect the feasible region of possible solutions?

Tip:

Always check if the constraint is nonlinear or linear, as this affects the complexity of the system of equations in the Lagrange multiplier method.