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The problem is about finding the break-even point for a company's production and sales of a product, where the cost and revenue functions are provided.

### Given:
- The total cost function \( C(x) = 5000 + 3x^2 - 2x \), where \( x \) represents the number of units produced.
- The total revenue function \( I(x) = 100x - 0.5x^2 \), where \( x \) represents the number of units sold.

To find the break-even point, we set the cost function equal to the revenue function, i.e., \( C(x) = I(x) \).

\[
5000 + 3x^2 - 2x = 100x - 0.5x^2
\]

Now, let's solve for \( x \). I'll handle the algebra and provide the break-even point.The solutions to the equation are complex numbers, indicating that there is no real break-even point where the cost and revenue functions intersect. This suggests that the company either doesn't break even or the given functions may result in losses or some other scenario. Therefore, none of the provided answers A, B, C, or D seem correct.

Would you like further clarification or a deeper analysis? 

Here are five related questions for exploration:
1. What does the absence of a real break-even point imply in this context?
2. How do cost functions and revenue functions typically behave in a real-world scenario?
3. How could the company adjust its production to achieve a break-even point?
4. What is the significance of the quadratic terms in the cost and revenue functions?
5. How can we interpret complex solutions in economic models?

**Tip:** Always verify if cost and revenue functions reflect realistic values for your situation to ensure the model's usefulness.

En una empresa de tecnología, se ha observado que el costo total de producción de un nuevo producto C(x) está dado por la expresión C(x)=5000+3x^2−2x, donde x representa la cantidad de unidades producidas. Por otro lado, el ingreso total I(x) generado por la venta de x unidades de este producto se calcula como I(x)=100x−0.5x^2. De acuerdo con esta información, ¿Cuál es el punto de equilibrio o la operación en dicho punto?

This problem involves finding the break-even point of a product using quadratic cost and revenue functions. The cost function is C(x)=5000+3x^2−2x, and the revenue function is I(x)=100x−0.5x^2. The solution involves equating the cost and revenue functions and solving the resulting quadratic equation. Suitable for high school students studying advanced algebra.

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