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### Solution:

The problem involves analyzing the graph of the derivative \( f'(x) \) of a function \( f(x) \). We are asked to determine at which points the original function \( f \) has a **relative maximum**.

#### **Key Observations:**

1. **Behavior of the derivative \( f'(x) \)**:
   - A **relative maximum** occurs at a point \( x = c \) if:
     - \( f'(c) = 0 \) (i.e., a horizontal tangent in \( f \)).
     - \( f'(x) \) changes **from positive to negative** at \( x = c \).
   
   This means we are looking for where the derivative crosses the x-axis (or touches it with a horizontal tangent) and the slope transitions from **positive to negative**.

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#### **Step-by-Step Analysis:**
Looking at the graph of \( f'(x) \):

- **At \( x = -2 \)**:
  - \( f'(-2) = 0 \) (it touches the x-axis).
  - **Sign change:** \( f'(x) \) changes from **positive to negative**.
  - Therefore, \( f(x) \) has a **relative maximum** at \( x = -2 \).

- **At \( x = 1 \)**:
  - \( f'(1) = 0 \) (horizontal tangent line).
  - **Sign change:** \( f'(x) \) changes from **positive to negative**.
  - Therefore, \( f(x) \) has a **relative maximum** at \( x = 1 \).

- **At \( x = 3 \)**:
  - \( f'(3) = 0 \) (horizontal tangent line).
  - **Sign change:** \( f'(x) \) changes from **positive to negative**.
  - Therefore, \( f(x) \) has a **relative maximum** at \( x = 3 \).

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#### **Answer:**
Since \( f(x) \) has relative maxima at \( x = -2 \), \( x = 1 \), and \( x = 3 \), the correct answer is:

**(E) \(-2\), 1, and 3**

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Do you need further clarifications or details on this?  
Here are five related questions you might find helpful:

1. How do you determine whether a critical point is a maximum or minimum using the first derivative test?
2. What does it mean when the derivative \( f'(x) \) touches but does not cross the x-axis?
3. Can a function have a maximum or minimum at a point where \( f'(x) \) is undefined?
4. What is the significance of the second derivative \( f''(x) \) in determining maxima and minima?
5. How does the behavior of the derivative affect the concavity of the original function?

**Tip:** Always check where the derivative changes signs to determine relative maxima and minima—it provides a clearer picture than relying on values alone!

The graph of the derivative of a function f is shown. The graph has horizontal tangent lines at x = -1, x = 1, and x = 3. At which of the following values of x does f have a relative maximum?

In this problem, we analyze the graph of the derivative of a function to determine where the function has relative maxima. We identify points at x = -2, 1, and 3 where the derivative changes sign from positive to negative, indicating relative maxima.

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