https://file.aiquickdraw.com/mathbot/file/1726361263880kxv4e6rx.jpg

Let's analyze the graph provided and answer the question step by step.

### The Problem:
We are asked for:
1. The value of \( f(-1) \).
2. The derivative \( f'(2) \).

### 1. Finding \( f(-1) \):
From the graph, we can see that \( f(-1) \) is the y-value when \( x = -1 \). Observing the graph:
- When \( x = -1 \), the function appears to intersect the y-axis at approximately \( y = 8 \).

Thus, \( f(-1) = 8 \).

### 2. Finding \( f'(2) \):
The derivative \( f'(2) \) represents the slope of the tangent line to the curve at \( x = 2 \). By analyzing the graph at this point:
- At \( x = 2 \), the function reaches a horizontal point (maximum). Therefore, the slope of the tangent line is zero at this point.

Hence, \( f'(2) = 0 \).

### Answer:
Given the information above:
- \( f(-1) = 8 \).
- \( f'(2) = 0 \).

The correct answer to the problem is **D: 0**.

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Would you like more details or have any questions? 

Here are some related questions to explore further:
1. How can you identify the derivative of a function from its graph?
2. What is the geometric interpretation of the derivative?
3. Why does the slope of the tangent line at a local maximum or minimum equal zero?
4. What are the implications of a function's concavity on its derivative?
5. How can you estimate higher-order derivatives from a graph?

**Tip:** When the tangent to a graph is horizontal (flat), the derivative at that point is always zero. This often occurs at local maxima or minima.

A partir de la gráfica de la función f mostrada en la figura, ¿cuál es el valor de f(-1). f'(2)?

Analyze the graph of a function to determine f(-1) and f'(2). Understand how to interpret the derivative at a specific point and the slope of a tangent line. Suitable for high school students in Grades 11-12.

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