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Below is a graph which has been divided into four sections. In which of these sections is the derivative of the function always negative?

To determine where the derivative of the function is always negative, we need to assess the behavior of the function's slope in each section. The derivative of a function represents its slope: a positive slope means the function is increasing, while a negative slope means the function is decreasing.

### Analyzing each section:
- **Section A (Leftmost section):** The curve is sloping downwards as it moves to the right, indicating that the derivative is negative.
- **Section B:** The function is increasing first, then decreasing. This means the derivative is positive at first and then becomes negative.
- **Section C:** The function decreases as it moves to the right, meaning the derivative is negative throughout this section.
- **Section D:** The function starts increasing, meaning the derivative is positive.

### Conclusion:
The derivative of the function is always negative in **Section A** and **Section C**.

Would you like further details or explanations? 

Here are 5 related questions:
1. What is the significance of a zero derivative in a function?
2. How can the second derivative help us understand concavity?
3. What are critical points, and how are they related to the derivative?
4. How does one find inflection points in a graph like this?
5. How can the behavior of the derivative tell us about local maxima or minima?

**Tip:** A function's derivative changes sign at local maxima and minima, which are key points for analyzing behavior.

Understand how to determine where the derivative of a function is negative by analyzing the slope of a graph in different sections. This example explores key calculus concepts and applies the derivative theorem. Suitable for students in Grades 10-12.

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