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Solve for f. Find the break-even points. Show. what first point x=, r(x) and c(x)=, second point x=, r(x) and c(x)=

I have reviewed the images you provided. They contain problems related to revenue \( R(x) \), cost \( C(x) \), and profit functions, as well as instructions to graph and find break-even points.

### Break-even points:
The break-even points occur where the revenue function \( R(x) \) equals the cost function \( C(x) \), i.e.,

\[
R(x) = C(x)
\]

### Given functions:
1. **Revenue function**: \( R(x) = 2800x - 60x^2 \) (in thousands of dollars)
2. **Cost function**: \( C(x) = 5000 + 500x \) (in thousands of dollars)

To find the break-even points, we need to solve the equation:

\[
2800x - 60x^2 = 5000 + 500x
\]

Let's solve this quadratic equation to find the values of \( x \) (the number of computers in thousands) that lead to break-even points.The break-even points for the given revenue and cost functions are:

\[
x_1 = \frac{115}{6} - \frac{5\sqrt{409}}{6} \quad \text{and} \quad x_2 = \frac{115}{6} + \frac{5\sqrt{409}}{6}
\]

Let's approximate these values to decimals for better understanding.The approximate break-even points (number of computers in thousands) are:

1. \( x_1 \approx 2.31 \) thousand computers
2. \( x_2 \approx 36.02 \) thousand computers

At these two points, the revenue equals the cost, meaning the company neither makes a profit nor incurs a loss.

Would you like to explore more details, or have any further questions? Here are some related questions:

1. How can we calculate the maximum revenue?
2. What is the profit function and how is it graphed?
3. How do we find the quantity of computers that maximizes profit?
4. What is the maximum profit achieved?
5. How do the break-even points affect business decisions?

**Tip:** Understanding break-even points helps in determining the range of production levels that are profitable.

This problem involves solving a quadratic equation to find the break-even points where the revenue function R(x) equals the cost function C(x). Learn how to analyze these financial functions and interpret their real-world meaning in a business context.

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