To optimize the function $z = 4x^2 + 3xy + 6y^2$ subject to the constraint $x + y = 56$, we can use the method of Lagrange multipliers.

The method of Lagrange multipliers involves introducing an auxiliary function called the Lagrangian, defined as:

$\mathcal{L}(x, y, \lambda) = 4x^2 + 3xy + 6y^2 + \lambda (56 - x - y)$

where $\lambda$ is the Lagrange multiplier.

To find the optimal points, we need to find the partial derivatives of $\mathcal{L}$ with respect to $x$, $y$, and $\lambda$ and set them to zero:

1. $\frac{\partial \mathcal{L}}{\partial x} = 8x + 3y - \lambda = 0$
2. $\frac{\partial \mathcal{L}}{\partial y} = 3x + 12y - \lambda = 0$
3. $\frac{\partial \mathcal{L}}{\partial \lambda} = 56 - x - y = 0$

Now, let's solve this system of equations:

1. $8x + 3y = \lambda$
2. $3x + 12y = \lambda$
3. $x + y = 56$

First, we equate the two expressions for $\lambda$:

$8x + 3y = 3x + 12y$

Solving for $x$ and $y$:

$5x = 9y$ $x = \frac{9}{5}y$

Substitute $x = \frac{9}{5}y$ into the constraint $x + y = 56$:

$\frac{9}{5}y + y = 56$ $\frac{14}{5}y = 56$ $y = 56 \cdot \frac{5}{14} = 20$

Now substitute $y = 20$ back into $x = \frac{9}{5}y$:

$x = \frac{9}{5} \cdot 20 = 36$

Thus, the optimal values of $x$ and $y$ are $x = 36$ and $y = 20$. To verify, substitute these values back into the constraint:

$36 + 20 = 56$

Finally, we calculate the optimized value of $z$:

$z = 4(36)^2 + 3(36)(20) + 6(20)^2$ $z = 4(1296) + 3(720) + 6(400)$ $z = 5184 + 2160 + 2400$ $z = 9744$

So, the optimized value of the function $z$ is 9744.

Would you like any more details or have any questions?

Here are some questions you might want to ask:

1. How does the method of Lagrange multipliers work?
2. Can you explain the steps to derive the Lagrangian function?
3. How do we solve systems of equations involving constraints?
4. What are the applications of optimization in real-world scenarios?
5. Can you solve another optimization problem with a different constraint?

Tip: When solving optimization problems, always double-check your constraint substitutions to ensure accuracy.