Suppose a company's revenue function is given by  and its cost function is given by , where  is thousands of units sold/produced, while  and  are in total dollars of revenue and cost, respectively.

A) Find a simplified expression for the profit function

 
Correct  

B) How many items (in thousands) need to be sold to maximize profits? (Round your answer to two decimal places.)

Answer: 
233.31
Correct thousand units must be sold.

C) Predict the maximum profit? (Round your answer to two decimal places.)

Answer: 
 dollars

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.

Suppose a company's revenue function is given by and its cost function is given by , where  is thousands of units sold/produced, while  and  are in total dollars of revenue and cost, respectively. A) Find a simplified expression for the profit function. B) How many items (in thousands) need to be sold to maximize profits? (Round your answer to two decimal places.) C) Predict the maximum profit? (Round your answer to two decimal places.)

Learn how to find a simplified profit function and determine the maximum profit point by using revenue and cost functions. Explore key optimization concepts such as the first and second derivative tests, with step-by-step solutions. Suitable for students in Grades 11-12 or introductory college-level courses.

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