To answer this, let's first establish the given information and solve each part systematically.

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Given Information

1. Revenue function: $R(x)$
2. Cost function: $C(x)$
3. $x$ represents thousands of units sold/produced.
4. Profit function, $P(x)$, is defined as the difference between revenue and cost: $P(x) = R(x) - C(x)$

Without the specific forms of $R(x)$ and $C(x)$, I'll illustrate a general solution process. You can share the exact functions if you need the specific calculations.

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Solution Outline

Part (A) – Finding the Profit Function, $P(x)$

The profit function is derived as: $P(x) = R(x) - C(x)$ Given specific forms of $R(x)$ and $C(x)$, simplify this expression to find a clearer profit function.

Part (B) – Maximizing Profit

To maximize profit, we:

1. Differentiate $P(x)$ with respect to $x$ to get $P'(x)$.
2. Set $P'(x) = 0$ and solve for $x$.
3. Use the second derivative test, $P''(x)$, to confirm a maximum.

You provided the answer, $x = 233.31$ thousand units, as the point that maximizes profit.

Part (C) – Maximum Profit Value

To find the maximum profit:

1. Substitute $x = 233.31$ into the profit function $P(x)$ (the result from Part A).
2. Compute $P(233.31)$ to get the maximum profit, rounded to two decimal places.

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Please provide the explicit forms of $R(x)$ and $C(x)$ for specific calculations. Let me know if you'd like the general solution in more detail!

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Questions for Practice:

1. How would you calculate maximum revenue if only the revenue function is given?
2. What other financial metrics can be derived from cost and revenue functions?
3. How does the shape of the profit function affect the maximum profit point?
4. How do marginal revenue and marginal cost relate to maximizing profit?
5. What role does the second derivative test play in profit maximization?

Tip:

When maximizing functions, ensure the value you find corresponds to a maximum by using the second derivative test or analyzing the function’s behavior around that point.