Let's solve the problem step by step. The graph shown is **NOT** the graph of \( f(x) \), but of \( f'(x) \) (the derivative of \( f(x) \)). --- ### **Step 1: Maximum of \( f(x) \)** **Question:** Where does the graph of \( f(x) \) have a maximum (if any)? #### **Explanation:** A maximum of \( f(x) \) occurs when \( f'(x) \) (the graph shown) changes from positive to negative (crosses the x-axis from above to below). - From the graph, \( f'(x) \) crosses the x-axis at \( x = -1 \). #### **Conclusion:** - **At \( x = -1 \), \( f(x) \) has a maximum.** --- ### **Step 2: Where is \( f(x) \) increasing?** **Question:** Where is the graph of \( f(x) \) increasing? #### **Explanation:** \( f(x) \) is increasing wherever \( f'(x) > \) (i.e., above the x-axis). - From the graph, \( f'(x) \) is above the x-axis between \( x = -3 \) and \( x = -1 \). #### **Conclusion:** - **\( f(x) \) is increasing on the interval \( (-3, -1) \).** --- ### **Final Answers:** **A.** The graph of \( f(x) \) has a maximum at \( x = -1 \). **B.** The graph of \( f(x) \) is increasing on the interval \( (-3, -1) \).

# Analysis of the Graph of \( f'(x) \) The graph provided represents the derivative \( f'(x) \) of a function \( f(x) \). We will analyze the graph to determine where \( f(x) \) has a maximum and where it is increasing. --- ## Step 1: Identifying Maximum of \( f(x) \) ### Question: Where does \( f(x) \) have a maximum? #### Explanation: A maximum of \( f(x) \) occurs when \( f'(x) \) transitions from positive to negative, indicating that the slope of \( f(x) \) is changing from increasing to decreasing. - From the graph, we see that \( f'(x) \) crosses the x-axis at \( x = -1 \). - Prior to \( x = -1 \), \( f'(x) \) is positive, and after \( x = -1 \), it is negative. #### Conclusion: - **At \( x = -1 \), \( f(x) \) has a maximum.** --- ## Step 2: Determining Where \( f(x) \) is Increasing ### Question: Where is \( f(x) \) increasing? #### Explanation: The function \( f(x) \) is increasing where \( f'(x) > 0 \) (i.e., where the graph of \( f'(x) \) is above the x-axis). - From the graph, \( f'(x) \) is above the x-axis on the interval \( (-3, -1) \). #### Conclusion: - **\( f(x) \) is increasing on the interval \( (-3, -1) \).** --- ## Final Answers: - **A.** The graph of \( f(x) \) has a maximum at \( x = -1 \). - **B.** The graph of \( f(x) \) is increasing on the interval \( (-3, -1) \).