Let's solve the problem step by step: ### Step 1: **Understand the Problem** We are given the graph of a function and asked to find the intervals where the function is increasing and decreasing. ### Step 2: **Identify Critical Points** Look for points where the function changes from increasing to decreasing or vice versa (these are usually at the "peaks" and "valleys" of the curve). From the graph: - The function reaches a **maximum** at \( x \approx -2 \) - The function reaches a **minimum** at \( x \approx 1 \) ### Step 3: **Determine Intervals** - **Increasing intervals**: Where the function goes up as \( x \) increases. - **Decreasing intervals**: Where the function goes down as \( x \) increases. #### From left to right: - The function **increases** from \( x = -\infty \) to \( x = -2 \). - It **decreases** from \( x = -2 \) to \( x = 1 \). - It **increases** again from \( x = 1 \) to \( x = \infty \). ### Step 4: **Write the Intervals** - **Increasing on the interval(s):** \( (-\infty, -2) \cup (1, \infty) \) - **Decreasing on the interval(s):** \( (-2, 1) \) --- ## **Final Answers:** **Increasing on the interval(s):** ``` (-\infty, -2) \cup (1, \infty) ``` **Decreasing on the interval(s):** ``` (-2, 1) ``` Let me know if you need further explanation!

Let's analyze the graph step-by-step in detail, focusing on the behavior of the function and identifying the intervals of increase and decrease. --- ### Step 1: Understanding the graph - The graph shows a continuous function with a clear maximum and minimum. - The key points are at: - A **maximum** point near \( x \approx -2 \) - A **minimum** point near \( x \approx 1 \) These points indicate where the function's slope changes from positive to negative or negative to positive. --- ### Step 2: Find critical points Critical points occur where the derivative \( f'(x) = 0 \) or is undefined. Since the graph is smooth and continuous: - At local maximum: \( x \approx -2 \) - At local minimum: \( x \approx 1 \) **Note:** Exact values are not given, but based on the shape, these are the points where the slope changes. --- ### Step 3: Determine increasing/decreasing intervals - **For \( x < -2 \):** - The graph is rising (slope positive). - Therefore, **\( f(x) \) is increasing** for \( x \in (-\infty, -2) \). - **Between \( x = -2 \) and \( x = 1 \):** - The graph descends from the maximum to the minimum. - The slope is negative here. - Therefore, **\( f(x) \) is decreasing** for \( x \in (-2, 1) \). - **For \( x > 1 \):** - The graph rises again after the minimum. - The slope is positive. - Therefore, **\( f(x) \) is increasing** for \( x \in (1, \infty) \). --- ### Step 4: Summarize in interval notation - **Increasing:** \( (-\infty, -2) \cup (1, \infty) \) - **Decreasing:** \( (-2, 1) \) --- ### **Final Answer:** > **The function is:** > > **Increasing on the intervals:** \(\boxed{(-\infty, -2) \cup (1, \infty)}\) > > **Decreasing on the interval:** \(\boxed{(-2, 1)}\) --- ### Additional notes: - The critical points are at \( x \approx -2 \) and \( x \approx 1 \). - The behavior matches the typical pattern: increase → maximum → decrease → minimum → increase. - Exact calculations are not possible without explicit function equations, but the qualitative analysis based on the graph suffices. If you need further detailed calculations involving derivatives or exact critical points, please provide the explicit function.