### Step 1: Analyze the Graph of \( f'(x) \) The given graph is of \( f'(x) \), which tells us about the **slope** of \( f(x) \), not the value of \( f(x) \) itself. We are also told that \( f(2) = 1 \). To find \( f(3) \), we need to consider how \( f(x) \) changes from \( x = 2 \) to \( x = 3 \). This change is given by the **area under the curve** of \( f'(x) \) between 2 and 3. **Explanation:** The derivative \( f'(x) \) gives the rate of change of \( f(x) \). The value of \( f(x) \) at a new point can be found by adding the net area under \( f'(x) \) from the starting point to the new point. --- ### Step 2: Estimate the Area Under \( f'(x) \) from 2 to 3 Look at the graph between \( x = 2 \) and \( x = 3 \): - In this interval, \( f'(x) \) is below the x-axis (negative values), which means \( f(x) \) is **decreasing**. - Since \( f(2) = 1 \) and the area under \( f'(x) \) from 2 to 3 is negative, \( f(3) \) will be **less than 1**. - But is \( f(3) \) negative? The area does not appear to be large enough to bring \( f(3) \) below zero (it looks much less than -1), so \( f(3) \) is likely still positive. **Explanation:** Since \( f'(x) \) is negative from 2 to 3, \( f(x) \) decreases but does not drop below zero if the area (magnitude) is less than 1. --- ### **Final Answer** **A. \( f(3) \) is positive.** We know \( f(3) \) is positive because \( f(2) = 1 \) and the negative area under \( f'(x) \) from 2 to 3 is not enough to make \( f(3) \) negative.