Determining Intervals Of Increase For F(x) = (1/3)x³ - 2x² + 3x Using Graphs
In this comprehensive guide, we will delve deep into the concept of intervals of increase for the function f(x) = (1/3)x³ - 2x² + 3x. This is a fundamental concept in calculus and understanding it is crucial for analyzing the behavior of functions. To fully grasp this, we'll leverage the power of graphical analysis and connect it to the underlying calculus principles. Our focus will be on how to determine the intervals where the function's value increases as x increases. This involves identifying sections of the graph where the curve slopes upwards. The question before us presents a graph of the function, and we aim to use this visual representation to pinpoint the specific intervals of increase. This is not just about finding the right answer but also understanding the 'why' behind it. We will break down the graph, looking for key features that indicate increasing behavior. This includes examining the slope of the curve at various points and identifying turning points, which mark transitions between increasing and decreasing intervals. The ability to interpret graphs and relate them to function behavior is a vital skill in mathematics, particularly in calculus and analysis. This skill extends beyond the classroom, finding applications in various fields such as physics, economics, and engineering, where understanding trends and changes is essential. Therefore, a thorough understanding of how to identify intervals of increase is an investment in a versatile and valuable analytical toolset. The challenge isn't just to read a graph, it's to interpret it, to understand the story it tells about the function's behavior. In the following sections, we'll dissect the function's graph, linking its visual features to the mathematical concept of increasing intervals, ensuring a clear and comprehensive understanding of this crucial aspect of function analysis.
Identifying Intervals of Increase: A Graphical Approach
To accurately determine the intervals of increase, we must first understand what an interval of increase truly signifies. In simple terms, an interval of increase is a section along the x-axis where the function's y-values are increasing as we move from left to right. Graphically, this corresponds to the portions of the curve that slope upwards. To pinpoint these intervals, we'll meticulously examine the provided graph of f(x) = (1/3)x³ - 2x² + 3x. Our methodology will involve several key steps, each designed to provide a clearer picture of the function's behavior. First, we'll scan the graph from left to right, paying close attention to the direction of the curve. We're looking for sections where the curve is ascending, indicating that the y-values are increasing as x increases. These upward-sloping sections are the visual indicators of our intervals of increase. Next, we'll identify the critical points on the graph, which are the points where the function changes direction – either from increasing to decreasing or vice versa. These points, also known as local maxima and minima, are crucial because they mark the boundaries of our intervals. They tell us where the increasing trend starts and stops. At these critical points, the tangent to the curve is horizontal, indicating a zero slope, which is a key concept in calculus. We'll then determine the x-coordinates of these critical points. These x-values are the endpoints of our intervals of increase. It's essential to accurately read these values from the graph, as they directly define the boundaries of our solution. Finally, we'll express the intervals of increase using interval notation. This involves using parentheses and brackets to indicate whether the endpoints are included or excluded from the interval. Parentheses are used for open intervals, where the endpoint is not included, while brackets are used for closed intervals, where the endpoint is included. In the context of intervals of increase, we typically use parentheses because the function is neither increasing nor decreasing at the critical points themselves. By following these steps methodically, we can confidently identify the intervals where the function's value increases, providing a solid understanding of its behavior. This graphical approach not only helps in solving this particular problem but also builds a strong foundation for more advanced calculus concepts.
Detailed Analysis of the Graph of f(x) = (1/3)x³ - 2x² + 3x
The graph of f(x) = (1/3)x³ - 2x² + 3x is a cubic function, characterized by its distinctive S-shape. This shape is crucial in understanding its behavior, including the intervals of increase. To accurately identify these intervals, we need to conduct a detailed analysis of the graph, focusing on key features such as turning points and the slope of the curve. Starting from the left, we observe that the graph initially slopes upwards, indicating that the function is increasing. This upward trend continues until it reaches a peak, a point where the function changes direction from increasing to decreasing. This peak represents a local maximum of the function. The x-coordinate of this local maximum is a critical value, marking the end of an interval of increase. As we move further along the x-axis, we see that the graph begins to descend after the peak, meaning the function is now decreasing. This downward slope persists until the graph reaches a trough, a low point where the function once again changes direction, this time from decreasing to increasing. This trough signifies a local minimum of the function, and its x-coordinate is another critical value, marking the start of a new interval of increase. After the trough, the graph starts to ascend again, indicating that the function is increasing once more. This upward trend continues indefinitely, as the graph extends towards positive infinity. The critical points, the local maximum and minimum, are essential in defining the intervals of increase. They act as boundaries, separating the sections where the function is increasing from those where it is decreasing. To determine the specific intervals, we need to accurately identify the x-coordinates of these critical points. By visually inspecting the graph, we can estimate these x-values. The local maximum appears to occur at x = 1, and the local minimum seems to be at x = 3. These values are crucial in defining our intervals of increase. Based on this analysis, we can conclude that the function is increasing in two distinct intervals: from negative infinity up to the local maximum at x = 1, and from the local minimum at x = 3 to positive infinity. This understanding of the graph's shape and critical points is fundamental to identifying intervals of increase. It highlights the importance of graphical analysis in understanding function behavior.
Determining the Intervals of Increase from the Graph
Based on our detailed analysis of the graph of f(x) = (1/3)x³ - 2x² + 3x, we can now confidently determine the intervals of increase. As we discussed, these intervals are the sections of the x-axis where the function's y-values increase as x increases, corresponding to the upward-sloping portions of the graph. Our analysis revealed two such intervals. The first interval of increase spans from negative infinity to the x-coordinate of the local maximum. We visually identified the local maximum as occurring at x = 1. Therefore, the first interval of increase is from negative infinity up to x = 1. In interval notation, this is represented as (-∞, 1). The parenthesis indicates that 1 is not included in the interval, as the function is neither increasing nor decreasing at the critical point itself. The second interval of increase begins at the x-coordinate of the local minimum and extends to positive infinity. We identified the local minimum as occurring at x = 3. Thus, the second interval of increase is from x = 3 to positive infinity. In interval notation, this is written as (3, ∞). Again, the parenthesis signifies that 3 is not included in the interval. Combining these two intervals, we find that the function f(x) = (1/3)x³ - 2x² + 3x is increasing on the intervals (-∞, 1) and (3, ∞). This means that as x moves from negative infinity towards 1, the function's value increases. Similarly, as x moves from 3 towards positive infinity, the function's value increases. These intervals are separated by the interval (1, 3), where the function is decreasing. This comprehensive determination of the intervals of increase is a direct result of our careful graphical analysis. By understanding the shape of the graph, identifying critical points, and applying interval notation, we have successfully pinpointed the regions where the function exhibits increasing behavior. This process underscores the power of graphical interpretation in understanding function dynamics.
Conclusion: The Intervals of Increase for f(x) = (1/3)x³ - 2x² + 3x
In conclusion, through our detailed exploration and analysis, we have successfully identified the intervals of increase for the function f(x) = (1/3)x³ - 2x² + 3x. By examining the graph of the function, we were able to visually determine the sections where the curve slopes upwards, indicating that the function's value increases as x increases. This graphical approach led us to identify two distinct intervals of increase. The first interval spans from negative infinity up to x = 1, the x-coordinate of the local maximum. In interval notation, this is represented as (-∞, 1). The second interval begins at x = 3, the x-coordinate of the local minimum, and extends to positive infinity. This is expressed in interval notation as (3, ∞). Therefore, the function f(x) = (1/3)x³ - 2x² + 3x is increasing on the intervals (-∞, 1) and (3, ∞). This means that as we move along the x-axis from the far left up to x = 1, the function's y-values are increasing. Similarly, as we continue along the x-axis from x = 3 towards the far right, the function's y-values increase again. The interval between x = 1 and x = 3 represents a region where the function is decreasing. Our analysis highlights the importance of graphical interpretation in understanding the behavior of functions. By visually examining the graph, we can quickly identify key features such as local maxima and minima, which are crucial in determining intervals of increase and decrease. This skill is essential in calculus and other areas of mathematics, as well as in various fields that utilize mathematical modeling and analysis. The ability to connect a function's graph to its underlying mathematical properties is a powerful tool for problem-solving and gaining insights into complex systems. This comprehensive understanding of intervals of increase not only provides a solution to the specific question at hand but also strengthens our broader mathematical reasoning and analytical skills. Thus, we've not just found an answer, but deepened our comprehension of function behavior.
Final Answer: The final answer is B. (-∞, 1) ∪ (3, ∞)
