Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, $R(x)$, and cost, $C(x)$, are in thousands of dollars, and $x$ is in thousands of units.
\[
R(x)=9 x-2 x^{2}, C(x)=x^{3}-5 x^{2}+2 x+1
\]
The production level for the maximum profit is about $\square$ units.
(Do not round until the final answer. Then round to the whole number as needed.)

Step 1 ：Define the profit function, \(P(x)\), as the difference between the revenue function, \(R(x)\), and the cost function, \(C(x)\). That is, \(P(x) = R(x) - C(x)\).

Step 2 ：Find the derivative of the profit function, \(P'(x)\), to find the critical points of the profit function.

Step 3 ：Set the derivative equal to zero and solve for \(x\) to find the critical points. The critical points are \(x = 1 - \sqrt{30}/3\) and \(x = 1 + \sqrt{30}/3\).

Step 4 ：Evaluate the profit function at these critical points and at the endpoints of the domain (if any) to find the maximum profit.

Step 5 ：The maximum profit is \(- (1 + \sqrt{30}/3)^3 + 6 + 7\sqrt{30}/3 + 3(1 + \sqrt{30}/3)^2\).

Step 6 ：The production level that yields the maximum profit is \(x = 1 + \sqrt{30}/3\).

Step 7 ：Final Answer: The production level for the maximum profit is about \(\boxed{3}\) units.