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.