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.

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.