Critical Numbers And Analysis Of F(x) = 4x³ - 9x
This article delves into the analysis of the function f(x) = 4x³ - 9x, focusing on determining its critical numbers and understanding its behavior. We will explore how to find these critical points and how they help us understand the function's increasing and decreasing intervals, as well as local extrema. To validate our analytical results, we'll also consider the use of graphing utilities. This comprehensive approach will provide a thorough understanding of the function's characteristics.
Determining Critical Numbers
Critical numbers are pivotal in understanding the behavior of a function. To find the critical numbers of f(x) = 4x³ - 9x, the first step involves calculating the derivative of the function, denoted as f'(x). The derivative represents the instantaneous rate of change of the function and is crucial for identifying points where the function's slope is zero or undefined. For our function, applying the power rule of differentiation, we get:
f'(x) = 12x² - 9
Critical numbers occur where the derivative is either equal to zero or undefined. In this case, f'(x) is a polynomial, and thus it is defined for all real numbers. Therefore, we only need to find the points where f'(x) = 0. Setting the derivative equal to zero, we have:
12x² - 9 = 0
To solve for x, we can first add 9 to both sides of the equation:
12x² = 9
Next, we divide both sides by 12:
x² = 9/12 = 3/4
Taking the square root of both sides, we get:
x = ±√(3/4) = ±√3 / 2
Thus, the critical numbers of the function f(x) = 4x³ - 9x are x = √3 / 2 and x = -√3 / 2. These points are crucial because they represent potential locations of local maxima, local minima, or points of inflection on the graph of the function. Understanding the nature of these critical points requires further analysis, such as examining the second derivative or creating a sign chart for the first derivative.
Analyzing Intervals of Increase and Decrease
After identifying the critical numbers, the next crucial step in analyzing the function f(x) = 4x³ - 9x is to determine the intervals where the function is increasing or decreasing. This analysis relies on examining the sign of the first derivative, f'(x) = 12x² - 9, in the intervals defined by the critical numbers. The critical numbers, x = -√3 / 2 and x = √3 / 2, divide the real number line into three intervals: (-∞, -√3 / 2), (-√3 / 2, √3 / 2), and (√3 / 2, ∞). By testing a value within each interval, we can determine the sign of f'(x) and, consequently, whether the function is increasing or decreasing.
In the interval (-∞, -√3 / 2), let's test x = -1. Plugging this value into f'(x), we get:
f'(-1) = 12(-1)² - 9 = 12 - 9 = 3
Since f'(-1) > 0, the function is increasing in the interval (-∞, -√3 / 2).
Next, consider the interval (-√3 / 2, √3 / 2). Let's test x = 0:
f'(0) = 12(0)² - 9 = -9
Here, f'(0) < 0, indicating that the function is decreasing in the interval (-√3 / 2, √3 / 2).
Finally, for the interval (√3 / 2, ∞), let's test x = 1:
f'(1) = 12(1)² - 9 = 12 - 9 = 3
As f'(1) > 0, the function is increasing in the interval (√3 / 2, ∞).
In summary, the function f(x) = 4x³ - 9x is increasing on the intervals (-∞, -√3 / 2) and (√3 / 2, ∞), and it is decreasing on the interval (-√3 / 2, √3 / 2). This information is crucial for sketching the graph of the function and identifying local extrema. The points where the function changes from increasing to decreasing or vice versa correspond to local maxima and minima, respectively.
Identifying Local Extrema
To pinpoint the local extrema of f(x) = 4x³ - 9x, we leverage the information obtained from the intervals of increase and decrease. Local extrema occur at critical points where the function changes its direction—from increasing to decreasing (local maximum) or from decreasing to increasing (local minimum). We've already identified the critical numbers as x = -√3 / 2 and x = √3 / 2. Now, we examine the function's behavior around these points.
We previously determined that f(x) is increasing on (-∞, -√3 / 2) and decreasing on (-√3 / 2, √3 / 2). This indicates that at x = -√3 / 2, the function transitions from increasing to decreasing, thus signifying a local maximum. To find the y-coordinate of this local maximum, we substitute x = -√3 / 2 into the original function:
f(-√3 / 2) = 4(-√3 / 2)³ - 9(-√3 / 2) = 4(-3√3 / 8) + 9√3 / 2 = -3√3 / 2 + 9√3 / 2 = 6√3 / 2 = 3√3
Thus, there is a local maximum at the point (-√3 / 2, 3√3).
Next, we consider the critical point x = √3 / 2. We know that f(x) is decreasing on (-√3 / 2, √3 / 2) and increasing on (√3 / 2, ∞). This change from decreasing to increasing implies a local minimum at x = √3 / 2. To find the corresponding y-coordinate, we substitute x = √3 / 2 into the function:
f(√3 / 2) = 4(√3 / 2)³ - 9(√3 / 2) = 4(3√3 / 8) - 9√3 / 2 = 3√3 / 2 - 9√3 / 2 = -6√3 / 2 = -3√3
Therefore, there is a local minimum at the point (√3 / 2, -3√3).
In summary, the function f(x) = 4x³ - 9x has a local maximum at (-√3 / 2, 3√3) and a local minimum at (√3 / 2, -3√3). These local extrema, along with the intervals of increase and decrease, provide a detailed understanding of the function's local behavior.
Verifying Results with a Graphing Utility
To ensure the accuracy of our analytical findings, employing a graphing utility is an invaluable step. Graphing utilities, such as Desmos or Wolfram Alpha, provide a visual representation of the function, allowing for a direct comparison between the calculated critical numbers, intervals of increase and decrease, and local extrema with the graphical depiction. For the function f(x) = 4x³ - 9x, we can use a graphing utility to plot the function and visually confirm our results.
Upon plotting the graph of f(x), we should observe the following:
- Critical Numbers: The graph should exhibit turning points at x = -√3 / 2 and x = √3 / 2, corresponding to the critical numbers we calculated. These points are where the tangent line to the curve is horizontal, indicating a zero slope.
- Intervals of Increase and Decrease: The graph should visually demonstrate the intervals where the function is increasing and decreasing. We expect the function to rise from left to right in the intervals (-∞, -√3 / 2) and (√3 / 2, ∞), and to fall from left to right in the interval (-√3 / 2, √3 / 2). This can be verified by observing the slope of the curve in each interval.
- Local Extrema: The graph should clearly show a local maximum at x = -√3 / 2 and a local minimum at x = √3 / 2. The y-coordinates of these points should match our calculated values of 3√3 and -3√3, respectively. The local maximum appears as a peak on the graph, and the local minimum appears as a trough.
By visually inspecting the graph, we can corroborate our analytical results. Any discrepancies between the calculated values and the graphical representation would warrant a review of our calculations or analysis. The graphing utility serves as a powerful tool for both verification and a deeper understanding of the function's behavior.
In conclusion, through analytical methods and graphical verification, we have thoroughly analyzed the function f(x) = 4x³ - 9x. We successfully determined the critical numbers, identified intervals of increase and decrease, and located local extrema. This comprehensive approach provides a complete understanding of the function's behavior and characteristics.
