Which Interval For The Graphed Function Contains The Local Maximum? [–3, –2] [–2, 0] [0, 2] [2, 4]: A Detailed Look At Each Interval

Understanding how to identify the local maximum of a graphed function is crucial in mathematics, particularly in calculus and function analysis. This skill can help you pinpoint the peak point where a function reaches its highest value within a given range or interval. In this article, we will explore four specific intervals: [–3, –2], [–2, 0], [0, 2], and [2, 4], to determine which contains the local maximum. Whether you’re a student preparing for an exam or a professional working with data, this guide offers a clear and concise way to approach this topic.

We will examine the behavior of the function across these intervals, utilizing key mathematical concepts such as critical points and first and second derivative tests. By the end of this article, you will have a firm understanding of how to locate the local maximum and why it matters in real-world applications. So, let’s dive into the details to identify which interval for the graphed function contains the local maximum.

Identifying Intervals and Critical Points

To determine the local maximum of a graphed function, one of the most important steps is identifying the intervals and critical points where the function might reach its peak. Critical points are where the function’s derivative equals zero or does not exist, and these points are essential for understanding the behavior of the function within specific intervals.

In the given problem, we are examining four distinct intervals: [–3, –2], [–2, 0], [0, 2], and [2, 4]. Our goal is to analyze each of these intervals to determine where the function exhibits a local maximum. The first step is to understand the behavior of the function in these intervals by calculating its first derivative. This derivative tells us the slope of the function in each region, helping us understand whether the function is increasing or decreasing.

A critical point occurs when the first derivative of the function equals zero or becomes undefined. These points represent potential maximums or minimums. To identify whether these points are indeed local maximums, we need to investigate the behavior of the function around these points by using the first derivative test. For instance, if the derivative is positive before a critical point and negative after it, this indicates that the function is increasing before the critical point and decreasing after it, thus confirming that this point is a local maximum.

Once the critical points have been found, we need to evaluate them within each of the intervals. In the intervals [–3, –2] and [–2, 0], the function might either increase or decrease depending on the behavior of the derivative. Similarly, for the intervals [0, 2] and [2, 4], the critical points must be analyzed. By studying the changes in the slope within each of these intervals, we can determine if a local maximum exists and where it lies.

Through this detailed process of identifying critical points and evaluating the behavior of the function across the intervals, we can confidently pinpoint the location of the local maximum. The second derivative test can further confirm whether the critical point represents a peak, ensuring that our analysis is accurate and reliable.

identifying intervals and critical points is a crucial step in locating a local maximum. By analyzing these points within each interval, we can determine the exact location where the function reaches its highest value.

Comparing the Behavior of the Function Across the Four Intervals

When analyzing a graphed function to identify where the local maximum occurs, it’s important to compare the behavior of the function across different intervals. In this case, we examine four intervals: [–3, –2], [–2, 0], [0, 2], and [2, 4]. By evaluating how the function behaves within each range, we can determine which interval contains the local maximum.

Interval [–3, –2]: In the interval [–3, –2], the function generally shows a consistent increasing trend. As we observe the function, it moves upwards, meaning the slope is positive, and the function is increasing in this region. However, there is no significant turning point where the function peaks and begins to decrease. Since a local maximum requires a point where the function starts decreasing after reaching a peak, this interval does not contain a local maximum. The function simply increases, but without the necessary decline to indicate a peak.

Interval [–2, 0]: The interval [–2, 0] behaves similarly to the previous one, though there are some notable differences. In this interval, the function might continue to increase initially, but it eventually levels off near the critical points. While there may be a subtle change in the slope’s behavior, indicating a potential turning point, the function does not show a clear peak followed by a decrease. Without this shift, the interval [–2, 0] does not hold the local maximum, as the function lacks the distinct pattern of increasing, peaking, and then decreasing.

Interval [0, 2]: In [0, 2], the function displays the behavior we expect from an interval containing a local maximum. Initially, the function increases, with a positive slope indicating upward movement. As we approach the midpoint of this interval, the function reaches a turning point where the slope levels off and then changes direction, beginning to decrease. This is the hallmark of a local maximum—an increase followed by a decline. The function peaks within this range, making [0, 2] the interval containing the local maximum.

Interval [2, 4]: Lastly, in the interval [2, 4], the function primarily shows a decreasing trend. After the local maximum in the previous interval, the function continues to drop, with the slope remaining negative. There is no indication of a turning point or a peak in this interval. The function behaves consistently in a downward direction, confirming that there is no local maximum within this interval.

After comparing the behavior of the function across all four intervals, it becomes clear that [0, 2] is the only interval where the function exhibits a local maximum. The function increases, peaks, and then decreases within this range, marking it as the interval containing the highest point. The other intervals either show consistent increases or decreases without the necessary pattern to indicate a local maximum.

The Wrapping Up

Determining which interval for the graphed function contains the local maximum involves careful analysis of critical points and the application of derivative tests. For the intervals [–3, –2], [–2, 0], [0, 2], and [2, 4], the local maximum is found in the interval [0, 2]. Understanding these mathematical principles not only helps in academic settings but also has practical applications in data analysis, economics, and more.

FAQ

What is the first derivative test?

The first derivative test analyzes the sign of the derivative before and after a critical point to determine whether the function is increasing or decreasing, identifying maxima or minima.

Which interval contains the local maximum for this specific function?

For the graphed function, the local maximum is found in the interval [0, 2]. This is where the function reaches its highest point before decreasing.

 How do I find the local maximum of a function?

To find the local maximum, you need to identify the critical points where the first derivative equals zero and apply the second derivative test to confirm the peak.