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The document you uploaded contains several mathematical problems related to derivatives, maxima, minima, and utility functions. Here’s a brief breakdown of what it includes:

1. **Derivatives**: Problems asking to find the derivative of various functions and simplify if possible. These include polynomial, exponential, logarithmic, and product rule applications.

2. **Differentiation exercises**: Problems that require finding the derivative of implicitly defined functions.

3. **Relative maxima and minima**: Problems that involve determining the relative maximum and minimum points of functions.

4. **Absolute maxima and minima**: These problems ask for the absolute maxima and minima of functions within a given interval.

5. **Application problems**: Two problems related to a company's utility function. These involve determining when the company had the highest and lowest utility (profit) and how much those values were.

Would you like assistance solving a specific problem from this document? If so, let me know which one, and I can walk you through the solution!

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Here are five follow-up questions to deepen understanding of the topics:
1. How do you apply the chain rule when differentiating composite functions?
2. What is the significance of finding critical points when solving for maxima and minima?
3. How can you determine whether a critical point is a local maximum or minimum?
4. What role do second derivatives play in curve sketching?
5. How do we approach optimization problems involving utility functions?

**Tip**: When solving for maxima or minima, always check the endpoints of the interval, especially for absolute extrema!

The document contains problems related to derivatives, maxima, minima, and utility functions. It asks to solve derivatives, determine relative and absolute extrema, and apply utility functions to find maximum and minimum values over given intervals.

Solve derivatives, identify relative and absolute extrema, and analyze utility functions in this detailed university-level mathematics problem set. Includes exercises on derivatives, maxima, and minima with real-world applications in utility functions.

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