Use the method of Lagrange multipliers to find the minimum value of
𝑓(𝑥, 𝑦) = 6𝑥
2 + 5𝑦
2 − 𝑥𝑦 subject to the constraint 2𝑥 + 𝑦 = 24

To find the minimum value of \( f(x, y) = 6x^2 + 5y^2 - xy \) subject to the constraint \( 2x + y = 24 \), we will use the method of Lagrange multipliers. The steps are as follows:

1. **Define the Lagrange function:**

   \[
   \mathcal{L}(x, y, \lambda) = f(x, y) + \lambda (g(x, y) - c)
   \]

   where \( f(x, y) = 6x^2 + 5y^2 - xy \) and the constraint \( g(x, y) = 2x + y - 24 \).

   Therefore, the Lagrange function is:

   \[
   \mathcal{L}(x, y, \lambda) = 6x^2 + 5y^2 - xy + \lambda (2x + y - 24)
   \]

2. **Compute the partial derivatives and set them equal to zero:**

   \[
   \frac{\partial \mathcal{L}}{\partial x} = 12x - y + 2\lambda = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial y} = 10y - x + \lambda = 0
   \]
   \[
   \frac{\partial \mathcal{L}}{\partial \lambda} = 2x + y - 24 = 0
   \]

3. **Solve the system of equations:**

   From the third equation:

   \[
   y = 24 - 2x
   \]

   Substitute \( y = 24 - 2x \) into the first and second equations:

   \[
   12x - (24 - 2x) + 2\lambda = 0
   \]
   \[
   10(24 - 2x) - x + \lambda = 0
   \]

   Simplify these equations:

   \[
   12x - 24 + 2x + 2\lambda = 0 \implies 14x - 24 + 2\lambda = 0 \implies 2\lambda = 24 - 14x \implies \lambda = 12 - 7x
   \]

   \[
   240 - 20x - x + \lambda = 0 \implies 240 - 21x + \lambda = 0 \implies \lambda = 21x - 240
   \]

   Set the two expressions for \(\lambda\) equal to each other:

   \[
   12 - 7x = 21x - 240 \implies 252 = 28x \implies x = 9
   \]

   Substitute \( x = 9 \) back into the constraint equation to find \( y \):

   \[
   y = 24 - 2(9) = 6
   \]

4. **Verify the solution:**

   Substitute \( x = 9 \) and \( y = 6 \) back into the original function \( f(x, y) \):

   \[
   f(9, 6) = 6(9^2) + 5(6^2) - (9 \cdot 6)
   \]
   \[
   f(9, 6) = 6(81) + 5(36) - 54
   \]
   \[
   f(9, 6) = 486 + 180 - 54
   \]
   \[
   f(9, 6) = 612
   \]

Therefore, the minimum value of the function \( f(x, y) = 6x^2 + 5y^2 - xy \) subject to the constraint \( 2x + y = 24 \) is **612**.

Would you like any more details or have any questions?

Here are some related questions you might find interesting:
1. How do Lagrange multipliers work in three variables?
2. Can Lagrange multipliers be used for inequality constraints?
3. How can we verify if the critical point found is a minimum or maximum?
4. What are some real-life applications of Lagrange multipliers?
5. How does the method of Lagrange multipliers compare to other optimization methods?

**Tip:** When dealing with optimization problems, it's crucial to carefully check the constraint conditions and ensure that the critical points satisfy these constraints.

Learn how to find the minimum value of the function f(x, y) = 6x^2 + 5y^2 - xy subject to the constraint 2x + y = 24 using the method of Lagrange multipliers. Detailed step-by-step solution provided.

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