Determining Turning Points Relative Minima And Maxima Of F(x) = (1/3)x³ - 2x² + 3x

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Navigating the realm of calculus and function analysis, identifying turning points on a graph is a fundamental skill. These points, where the function transitions from increasing to decreasing or vice versa, provide critical insights into the function's behavior. In this comprehensive exploration, we will delve into the process of pinpointing turning points on the graph of the cubic function f(x) = (1/3)x³ - 2x² + 3x. We'll analyze the provided options, discuss the concepts of relative minima and maxima, and ultimately arrive at the accurate determination of the turning points.

Understanding Turning Points: Relative Minima and Maxima

Before diving into the specifics of our function, let's establish a firm understanding of what turning points represent. In calculus, turning points, also known as local extrema, are points on the graph of a function where the derivative changes sign. This change in sign signifies a shift in the function's direction – from increasing to decreasing or from decreasing to increasing.

  • Relative Minimum: A relative minimum, also referred to as a local minimum, is a point where the function's value is lower than the values at all nearby points. Imagine a valley in the graph; the bottom of the valley represents a relative minimum. The function decreases as it approaches the minimum point and increases as it moves away from it.
  • Relative Maximum: Conversely, a relative maximum, or local maximum, is a point where the function's value is higher than the values at all nearby points. Think of a peak on the graph; the top of the peak is a relative maximum. The function increases as it approaches the maximum point and decreases as it moves away from it.

The turning points are crucial for understanding the shape and behavior of a function's graph. They help us identify intervals where the function is increasing or decreasing, and they provide valuable information for optimization problems, where we aim to find the maximum or minimum value of a function within a given range.

Analyzing the Graph of f(x) = (1/3)x³ - 2x² + 3x

Now, let's turn our attention to the specific function in question: f(x) = (1/3)x³ - 2x² + 3x. This is a cubic function, characterized by its highest power of x being 3. Cubic functions typically exhibit a distinctive S-shaped curve, which means they often have two turning points: one relative minimum and one relative maximum. The graph mentioned in the prompt visually represents this curve, allowing us to identify these turning points.

To accurately determine the turning points from the graph, we need to carefully observe where the function changes its direction. Look for points where the curve transitions from going downwards (decreasing) to upwards (increasing), and vice versa. These transition points are the turning points.

The given options provide potential coordinates for these turning points. We need to visually compare these coordinates with the graph to determine which option accurately reflects the locations of the relative minimum and maximum.

Evaluating the Provided Options

Let's examine the options presented:

Option A: Relative minimum: (1, 1.333); Relative maximum: (3, 0)

This option suggests a relative minimum at the point (1, 1.333) and a relative maximum at the point (3, 0). To assess this, we would visually locate these points on the graph. We need to determine if the graph indeed has a low point (relative minimum) near x = 1 and a high point (relative maximum) near x = 3.

Option B: Relative minimum: (0, 0); Relative maximum: [Incomplete Option]

This option proposes a relative minimum at (0, 0). This is a crucial point to examine because (0,0) is the origin, and we can quickly verify if the graph has a minimum point there. The option is incomplete regarding the relative maximum, which means we cannot fully evaluate it without the complete information.

To accurately determine the correct answer, a visual inspection of the graph is essential. Without the actual graph to reference here, we can discuss the process of how one would do it and the mathematical reasoning behind it.

The Process of Graphically Determining Turning Points

When you have the graph in front of you, here's the step-by-step process to identify the turning points:

  1. Scan the Graph: Begin by scanning the entire graph to get a general sense of its shape and behavior. Look for the characteristic S-curve of a cubic function.
  2. Identify Potential Turning Points: Locate points where the graph changes direction. These will appear as peaks (potential relative maxima) and valleys (potential relative minima).
  3. Estimate Coordinates: For each potential turning point, estimate its x and y coordinates. These estimations should be as accurate as possible by referencing the graph's axes.
  4. Compare with Options: Compare your estimated coordinates with the options provided. The option that most closely matches your estimations is likely the correct one.
  5. Confirm Minimum vs. Maximum: Once you've identified a potential turning point, confirm whether it's a relative minimum or maximum. A relative minimum will be at the bottom of a valley, while a relative maximum will be at the top of a peak.

Mathematical Verification (Without the Graph)

While we've focused on graphically determining turning points, it's important to note that calculus provides a powerful tool for mathematically verifying these points. The process involves finding the derivative of the function, setting it equal to zero, and solving for x. The solutions for x represent the x-coordinates of the critical points, which are potential turning points. To confirm whether a critical point is a minimum or maximum, we can use the second derivative test.

  1. Find the First Derivative: Calculate the first derivative of the function, f'(x). This derivative represents the slope of the tangent line to the graph at any point.
  2. Set the Derivative to Zero: Set f'(x) = 0 and solve for x. The solutions are the critical points.
  3. Find the Second Derivative: Calculate the second derivative of the function, f''(x). This derivative tells us about the concavity of the graph.
  4. Apply the Second Derivative Test: Evaluate f''(x) at each critical point:
    • If f''(x) > 0, the critical point is a relative minimum.
    • If f''(x) < 0, the critical point is a relative maximum.
    • If f''(x) = 0, the test is inconclusive, and further analysis is needed.

For our function, f(x) = (1/3)x³ - 2x² + 3x, let's perform these steps:

  • First Derivative: f'(x) = x² - 4x + 3
  • Set to Zero: x² - 4x + 3 = 0. Factoring, we get (x - 1)(x - 3) = 0. Thus, x = 1 and x = 3 are the critical points.
  • Second Derivative: f''(x) = 2x - 4
  • Second Derivative Test:
    • f''(1) = 2(1) - 4 = -2. Since f''(1) < 0, x = 1 corresponds to a relative maximum.
    • f''(3) = 2(3) - 4 = 2. Since f''(3) > 0, x = 3 corresponds to a relative minimum.

To find the y-coordinates of the turning points, we substitute x = 1 and x = 3 back into the original function:

  • f(1) = (1/3)(1)³ - 2(1)² + 3(1) = 1/3 - 2 + 3 = 4/3 = 1.333
  • f(3) = (1/3)(3)³ - 2(3)² + 3(3) = 9 - 18 + 9 = 0

Therefore, the relative maximum is at (1, 1.333) and the relative minimum is at (3, 0).

Conclusion

Based on our mathematical analysis, the turning points of the function f(x) = (1/3)x³ - 2x² + 3x are a relative maximum at (1, 1.333) and a relative minimum at (3, 0). This aligns with option A, which we can confirm is the correct answer if the graph visually supports these coordinates as a high point and a low point, respectively. This exercise highlights the power of combining graphical analysis with calculus techniques to fully understand the behavior of functions. The ability to identify and interpret turning points is a valuable asset in mathematics and its applications.