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The document you provided contains a practical case study regarding the analysis of a company's profit function. The function provided is:

\[
B(x) = \frac{x^2 - 32}{x + 10}
\]

where \( x \) represents the number of products sold. The goal is to analyze this function by determining the following:

1. **Domain and Range**: The domain excludes values that make the denominator zero, i.e., \( x = -10 \), and the range depends on the behavior of the function as \( x \to \infty \) and near the vertical asymptote.
2. **Asymptotes**: The vertical asymptote occurs where the denominator is zero, and the oblique asymptote comes from performing polynomial long division.
3. **Growth and Decline Intervals**: Use the derivative to determine where the function is increasing or decreasing.
4. **Critical Points**: These include the maxima or minima found using the first and second derivative tests.
5. **Concavity and Points of Inflection**: Analyze the second derivative to identify concavity and inflection points.

Would you like to begin solving one of these steps, or shall I go through all the details?

Here are some related questions to extend your understanding:

1. What is the significance of the vertical asymptote in the context of the profit function?
2. How do we use the first derivative to find the intervals of increase and decrease?
3. What are the steps to classify a critical point as a maximum or minimum?
4. How can we determine the oblique asymptote of the given function?
5. Why is it important to analyze concavity and points of inflection in business applications?

**Tip**: To determine vertical asymptotes, always look for values that make the denominator of a rational function zero, as this often indicates where the function is undefined.

The user submitted a problem involving the analysis of a company's profit function: B(x) = (x^2 - 32) / (x + 10), where x represents the number of products sold. The task is to determine the domain and range, calculate vertical and oblique asymptotes, find growth and decline intervals, identify critical points, and analyze concavity and inflection points.

This analysis covers the profit function B(x) = (x^2 - 32) / (x + 10), exploring its domain, asymptotes, intervals of growth and decline, critical points, and concavity. Suitable for calculus students, this problem offers a practical approach to understanding rational functions and optimization.

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