To solve the problem using the method of substitution, we'll follow these steps:

We are given the objective function and the constraint:

Objective function:
$z = 6x - 3x^2 + 2y$

Constraint:
$y - x^2 = 2$

Step 1: Solve the constraint for $y$

We start by solving the constraint equation for $y$:

$y = x^2 + 2$

Step 2: Substitute $y$ into the objective function

Now, substitute $y = x^2 + 2$ into the objective function $z$:

$z = 6x - 3x^2 + 2(x^2 + 2)$

Simplify this expression:

$z = 6x - 3x^2 + 2x^2 + 4$

$z = 6x - x^2 + 4$

Step 3: Take the derivative of $z$ with respect to $x$

To maximize $z$, we need to find its critical points. To do this, we first take the derivative of $z$ with respect to $x$:

$\frac{dz}{dx} = 6 - 2x$

Step 4: Set the derivative equal to zero to find critical points

Set the derivative equal to zero to find the values of $x$ that maximize $z$:

$6 - 2x = 0$

Solving for $x$:

$2x = 6$

$x = 3$

Step 5: Find the corresponding value of $y$

Substitute $x = 3$ into the constraint $y = x^2 + 2$ to find the corresponding value of $y$:

$y = 3^2 + 2 = 9 + 2 = 11$

Step 6: Calculate the maximum value of $z$

Now substitute $x = 3$ and $y = 11$ into the objective function $z = 6x - 3x^2 + 2y$ to calculate the maximum value of $z$:

$z = 6(3) - 3(3^2) + 2(11)$

$z = 18 - 3(9) + 22$

$z = 18 - 27 + 22 = 13$

Conclusion

The maximum value of $z$ is $\boxed{13}$, and it occurs when $x = 3$ and $y = 11$.

Would you like further details or have any questions about this solution?

Here are some related questions you might find useful:

1. How do you determine whether a critical point corresponds to a maximum or minimum?
2. What is the second derivative test, and how can it be used to verify the nature of the critical point?
3. How does the method of Lagrange multipliers differ from the method of substitution in constrained optimization problems?
4. How would you solve a similar problem if the constraint was linear, like $y = 2x + 1$?
5. What would happen to the solution if the objective function included a cubic term, such as $z = 6x - 3x^2 + 2x^3 + 2y$?

Tip: Always check the second derivative (or test values around critical points) to confirm whether a solution is a maximum or a minimum when performing optimization.