To minimize the function $f(x, y) = 48x + 96y - x^2 - 2xy - 9y^2$ subject to the constraint $20x + 4y = 216$, we will use the method of Lagrange multipliers. Here’s how:

1. Define the Lagrange Function:

   Given: $f(x, y) = 48x + 96y - x^2 - 2xy - 9y^2$ Constraint: $g(x, y) = 20x + 4y - 216 = 0$

   We introduce a Lagrange multiplier $\lambda$ and form the Lagrange function $\mathcal{L}(x, y, \lambda)$: $\mathcal{L}(x, y, \lambda) = f(x, y) + \lambda (216 - 20x - 4y)$

2. Compute Partial Derivatives:

   We need to set the partial derivatives of $\mathcal{L}$ with respect to $x$, $y$, and $\lambda$ equal to zero.

   Partial derivative with respect to $x$: $\frac{\partial \mathcal{L}}{\partial x} = 48 - 2x - 2y - 20\lambda = 0$

   Partial derivative with respect to $y$: $\frac{\partial \mathcal{L}}{\partial y} = 96 - 2x - 18y - 4\lambda = 0$

   Partial derivative with respect to $\lambda$: $\frac{\partial \mathcal{L}}{\partial \lambda} = 216 - 20x - 4y = 0$

3. Solve the System of Equations:

   We now have three equations: $48 - 2x - 2y - 20\lambda = 0$ $96 - 2x - 18y - 4\lambda = 0$ $216 - 20x - 4y = 0$

   Solving this system will give us the values of $x$, $y$, and $\lambda$ that minimize $f(x, y)$ subject to the constraint.

Let's proceed by solving these equations step-by-step.The solution to the system is: $x = 10, \quad y = 4, \quad \lambda = 1$

Thus, the values of $x = 10$ and $y = 4$ minimize the function $f(x, y)$ subject to the constraint $20x + 4y = 216$.

Would you like a detailed step-by-step explanation, or do you have any questions about this solution?

Here are five questions to expand your understanding:

1. How does the method of Lagrange multipliers work in optimization problems?
2. Why is it important to set partial derivatives to zero in this method?
3. What would change if the constraint equation had a different form, such as a nonlinear equation?
4. How would the approach differ if we were maximizing $f(x, y)$ instead of minimizing?
5. Can Lagrange multipliers be used for functions with more than two variables or constraints?

Tip: When solving optimization problems with constraints, always check that the solutions satisfy both the constraint and any boundaries of the problem's domain if they exist.