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Part 2 of your document involves solving the following constrained optimization problem:

### Given:

**Objective function:**
\[
f(x, y) = 3x^2 + xy + 2y^2
\]
**Subject to constraints:**
\[
-0.5x + y = 1 \quad \text{and} \quad x^2 - y - 5x + 7 = 0
\]

To solve this optimization problem, we will apply the **Lagrange multiplier** method. The Lagrange function is given by:
\[
\mathcal{L}(x, y, \lambda_1, \lambda_2) = 3x^2 + xy + 2y^2 + \lambda_1(-0.5x + y - 1) + \lambda_2(x^2 - y - 5x + 7)
\]
where \(\lambda_1\) and \(\lambda_2\) are the Lagrange multipliers corresponding to the constraints.

### Steps to solve:

1. **Set up the Lagrange system:**
   We need to take partial derivatives of \(\mathcal{L}\) with respect to \(x\), \(y\), \(\lambda_1\), and \(\lambda_2\) and set them equal to zero.

   \[
   \frac{\partial \mathcal{L}}{\partial x} = 6x + y - 0.5\lambda_1 + (2x - 5)\lambda_2 = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial y} = x + 4y + \lambda_1 - \lambda_2 = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda_1} = -0.5x + y - 1 = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda_2} = x^2 - y - 5x + 7 = 0
   \]

2. **Solve the system:**
   Solve the above system of equations to find the values of \(x\), \(y\), \(\lambda_1\), and \(\lambda_2\). This step can be done using MATLAB or another symbolic solver.

3. **Verify minimum condition:**
   Once the candidate values of \(x\) and \(y\) are found, we check whether they correspond to a minimum by computing the **Hessian matrix** of the Lagrange function and checking if it is positive definite at the solution point.

4. **Find the function value at the constrained minimum:**
   After confirming that the solution represents a minimum, calculate the value of the objective function \(f(x, y)\) at the optimal values of \(x\) and \(y\).

5. **MATLAB visualization:**
   Create a **contour plot** of the objective function with the constraints superimposed, and highlight the location of the constrained minimum.

Would you like help with a specific part of this process, such as solving the system of equations or plotting the results in MATLAB? Let me know!

Here are some questions to consider:
1. Would you like a detailed solution to the Lagrange multiplier setup?
2. Do you want help solving the system using MATLAB or another tool?
3. Should I explain how to verify the Hessian for a minimum?
4. Would you like assistance creating a contour plot in MATLAB?
5. Do you need clarification on the constraint handling method?

**Tip:** The Lagrange multiplier method is a powerful tool for solving constrained optimization problems, particularly when the constraints are non-linear like in this case.

Learn how to solve a constrained optimization problem using Lagrange multipliers. This tutorial covers solving a quadratic objective function with nonlinear constraints, using Jacobian and Hessian matrices to verify minimum values, and plotting the results in MATLAB.

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