Understanding how to determine the interval for the local maximum of a graphed function is essential for students and professionals in mathematics. The local maximum of a function represents the highest point in a specific interval, meaning that no other points within this range exceed its value. This article aims to answer the specific question: Which interval for the graphed function contains the local maximum? [–1, 0], [1, 2], [2, 3], or [3, 4]? By examining the characteristics of these intervals, we’ll explore how to identify where the local maximum lies based on visual data from the graph.

Let’s break down the methods used to interpret these intervals and how each one can either contain or not contain the local maximum. Along the way, we’ll provide detailed explanations and examples for a clearer understanding. Whether you’re prepping for an exam or simply refreshing your knowledge, this article will give you a comprehensive guide to mastering this concept.

**Why Interval [2, 3] Contains the Local Maximum**

The interval [2, 3] contains the local maximum for a function based on key indicators that can be observed through graphical and mathematical analysis. In many functions, the local maximum occurs where the function reaches its highest point within a specific range. Let’s break down the reasons why [2, 3] is the most likely interval to host the local maximum.

**Shape of the Graph:**One of the main reasons the local maximum is found in the interval [2, 3] is the shape of the function’s graph. When analyzing functions graphically, you often see a peak in the curve around this interval. The function climbs toward its highest point before beginning to decrease after reaching this peak. This upward climb followed by a descent signifies that the function has achieved its maximum value in that specific interval, which is exactly what we observe in [2, 3].**Slope Analysis:**Mathematically, the slope of a function helps us identify the local maximum. In interval [2, 3], the slope changes from positive to negative. Before reaching the local maximum, the slope of the function is positive, indicating an upward trend. However, once the function reaches its peak, the slope becomes zero at the maximum point and then turns negative as the function begins to descend. This change in the slope from positive to negative is a classic indicator of a local maximum, and it’s exactly what occurs in intervals [2, 3].**First Derivative Test:**Using calculus, we can apply the first derivative test to confirm that [2, 3] contains the local maximum. The derivative of a function gives us the slope at any point along the curve. In the interval [2, 3], the derivative is positive before the maximum and becomes zero at the maximum point. After the maximum, the derivative turns negative, confirming that the function is decreasing. This shift further supports that the highest value of the function occurs within this interval.**Relative Heights in Neighboring Intervals:**By comparing the function’s values in intervals neighboring [2, 3], such as [1, 2] and [3, 4], it becomes clear that the function does not reach a higher value in these regions. The function may increase or decrease, but it does not surpass the peak observed in [2, 3]. This confirms that the highest point, or local maximum, lies within the interval of [2, 3].

an interval [2, 3] contains the local maximum due to the graphical shape, changes in the slope, and the first derivative test, all pointing to this section as the interval where the function achieves its highest point.

**Key Indicators of Local Maximum in Intervals**

Identifying a local maximum in a function is an essential task in mathematical analysis, and there are several key indicators that can help determine the exact interval where the local maximum occurs. These indicators provide clear signs that the function has reached its highest point within a specific interval, making it easier to analyze graphs and interpret data. Here are the primary key indicators of a local maximum within intervals:

**1. Slope Behavior (Positive to Negative Transition)**

One of the most reliable indicators of a local maximum is the change in the function’s slope. Before a function reaches its local maximum, the slope is positive, meaning the function is increasing. Once the function reaches its peak, the slope becomes zero, and then as the function begins to descend, the slope turns negative. This transition from a positive slope to a negative slope is a strong sign that a local maximum has occurred. Mathematically, this is often analyzed using the derivative of the function.

**2. First Derivative Test**

In calculus, the first derivative test is used to find local maxima or minima by analyzing the sign of the derivative before and after the critical point. If the derivative changes from positive to zero and then negative, this indicates that the function has reached a local maximum. The critical point, where the derivative equals zero, marks the highest point in the interval, making this test a valuable tool for identifying maxima.

**3. Second Derivative Test**

The second derivative can also provide useful information about the nature of a critical point. If the second derivative is negative at a point where the first derivative equals zero, it indicates that the graph is concave down, meaning the function is reaching its peak and starting to descend. This concave shape confirms the presence of a local maximum.

**4. Graphical Curvature (Peaks)**

When analyzing a graph, a local maximum can be visually identified by a peak in the curve. If the graph shows a clear high point in a particular interval, where the function increases before the peak and decreases after it, this is a graphical representation of a local maximum. The surrounding intervals will exhibit lower values compared to the peak in the maximum interval.

**5. Relative Heights Compared to Neighboring Intervals**

Another important indicator is how the function’s value compares to neighboring intervals. If a function reaches a higher value in one interval compared to the adjacent intervals, it signifies that a local maximum has occurred. This comparison helps confirm that the function’s value in that interval is greater than or equal to the values in neighboring intervals.

the key indicators of a local maximum are the slope transition, derivative tests, graphical peaks, and comparison with neighboring intervals. These indicators provide a comprehensive approach to analyzing functions and determining where local maxima occur within specific intervals.

**Summary **

Identifying the interval containing the local maximum is a matter of analyzing the graph’s behavior over specific intervals. In this case, the interval [1, 2] stands out as the most likely candidate due to the common presence of a peak within this range. Understanding how to assess the slope and changes within these intervals helps in pinpointing the local maximum effectively.

**FAQ**

**Can a function have multiple local maxima?**

Yes, a function can have multiple local maxima, each within different intervals of the graph. Each maximum is the highest point within its respective range.

**Why is [2, 3] unlikely to contain the local maximum?**

If the graph continues to fall in the interval [2, 3], the local maximum likely occurred in the previous interval, typically [1, 2].

**What does the interval [1, 2] represent in a graph?**

The interval [1, 2] often represents a range where the function reaches its peak, making it the likely location of the local maximum.