To find the correct graph of the derivative $f'$, let's first analyze the original function $f(x)$ shown in the image.

Characteristics of the function $f(x)$:

• $f(x)$ is a piecewise linear function with sharp turns at $x = -2, 0, 2$.
• The slope of $f(x)$ in different segments:
  • From $x = -2$ to $x = -1$, the function is increasing, which means $f'(x)$ is positive.
  • From $x = -1$ to $x = 1$, the function is constant, which means $f'(x) = 0$.
  • From $x = 1$ to $x = 2$, the function is decreasing, which means $f'(x)$ is negative.

Derivative analysis:

1. Interval $(-2, -1)$: The slope of $f(x)$ is positive, so $f'(x)$ will be a positive constant in this region.
2. Interval $(-1, 1)$: The function is flat (constant), so the derivative $f'(x) = 0$ in this region.
3. Interval $(1, 2)$: The function is decreasing, so $f'(x)$ will be a negative constant in this region.

Step-by-Step Matching with the Options:

• Option 1: The graph shows a positive value for $f'(x)$ on the interval $(-2, -1)$, zero on the interval $(-1, 1)$, and negative on the interval $(1, 2)$, which matches the behavior of the derivative. This is the correct graph of $f'(x)$.

Thus, the correct option is Option 1.

Would you like a more detailed breakdown of how the derivative was calculated? Let me know if you have any questions or need more details.

Related questions:

1. How is the derivative of a piecewise linear function defined at sharp turns?
2. What are the key differences between a graph of a function and its derivative?
3. How can the slopes of linear segments be used to find the graph of the derivative?
4. Why does the derivative equal zero in regions where the function is constant?
5. What would the second derivative of this function look like?

Tip:

When analyzing the graph of a function and its derivative, always focus on the slope of the original function at each interval. Positive slopes yield positive derivative values, flat regions give zero, and negative slopes give negative derivative values.