To find the value of $x$ that maximizes the profit and the corresponding maximum profit, we need a clear understanding of the profit function and its critical points. If you provide the profit function or any relevant equations, we can analyze it in detail.

Based on the given information, here is how we might approach:

1. Critical Points: If $x = 3, 6, 9$ are mentioned, these could be critical points of the profit function, found by solving $P'(x) = 0$, where $P(x)$ is the profit function.

2. Evaluation: We substitute these critical points into the profit function $P(x)$ to determine the maximum profit.

3. Interpretation: From the summary provided:

   • Maximum Profit $= \$627$, which likely corresponds to one of these critical points, likely $x = 9$.

To proceed:

• Do you have a formula or further details on the profit function $P(x)$?
• Would you like a general explanation of how to optimize profit mathematically?

Next Steps:

1. Can you provide more details or clarify if you need step-by-step solutions for critical points?
2. Should I assume that $x = 9$ indeed gives $P(x) = 627$?
3. Would you like an explanation of how to confirm maximum profit using second derivatives or other criteria?
4. Should I illustrate how profit optimization works using a hypothetical function?
5. Do you have questions about why $x = 3, 6,$ and $9$ are considered?

Tip: Maximizing profit often involves analyzing derivatives. Setting $P'(x) = 0$ identifies critical points, while the second derivative test confirms their nature (maxima or minima).