Finding The Critical Number And Intervals For F(x) = 6(x-4)^(2/3)
In the realm of calculus, understanding the behavior of functions is paramount. One crucial aspect of function analysis is identifying critical numbers, which serve as pivotal points in determining intervals of increasing and decreasing behavior. This article delves into the function f(x) = 6(x-4)^(2/3), a seemingly simple yet insightful example that illuminates the significance of critical numbers. We will meticulously examine the function, locate its critical number, and discuss the implications for the function's overall behavior. The exercise of finding critical numbers is not merely an academic pursuit; it has profound practical applications in optimization problems across various fields, such as engineering, economics, and computer science. Understanding how to find and interpret critical numbers empowers us to make informed decisions and optimize outcomes in diverse real-world scenarios. So, let's embark on this journey of mathematical exploration and uncover the secrets hidden within the function f(x) = 6(x-4)^(2/3).
To determine the critical number of the function f(x) = 6(x-4)^(2/3), we must first understand what a critical number is. A critical number of a function is a point in the domain of the function where either the derivative of the function is zero or the derivative does not exist. These points are crucial because they often mark potential locations of local maxima, local minima, or points of inflection. To find the critical number A for our function, we need to follow a systematic approach. First, we compute the derivative of f(x) using the power rule and chain rule. Second, we identify the values of x where the derivative is either equal to zero or undefined. These values will be our critical numbers. The process involves algebraic manipulation and careful consideration of the domain of both the function and its derivative. By meticulously executing these steps, we can pinpoint the critical number A and pave the way for further analysis of the function's behavior.
Step-by-Step Calculation of the Derivative
The heart of finding critical numbers lies in computing the derivative of the function. For f(x) = 6(x-4)^(2/3), we employ the power rule and the chain rule. The power rule states that the derivative of x^n is nx^(n-1), and the chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x). Applying these rules, we first rewrite the function to emphasize the power and the inner function. Then, we apply the power rule to the outer function and the chain rule to the inner function (x-4). This yields an expression that needs simplification. We carefully combine terms and express the derivative in a form that is easy to analyze. The simplified form will reveal the points where the derivative is zero or undefined, which are the candidates for critical numbers. This step is crucial, as an incorrect derivative will lead to erroneous results. Therefore, we must double-check our calculations and ensure accuracy in applying the differentiation rules. The derivative, once correctly computed and simplified, will be the key to unlocking the critical number A and understanding the function's behavior.
Determining Where the Derivative is Zero or Undefined
Once we have the derivative of f(x), the next crucial step is to identify the values of x where the derivative is either zero or undefined. These points are the potential critical numbers of the function. A derivative is zero when the numerator of the derivative expression is zero, provided the denominator is not simultaneously zero. A derivative is undefined when the denominator of the derivative expression is zero, as division by zero is undefined. Therefore, we set the numerator of our derivative equal to zero and solve for x. This will give us the points where the function has a horizontal tangent. Next, we set the denominator equal to zero and solve for x. This will give us the points where the function has a vertical tangent or a cusp, or where the derivative simply does not exist. It is crucial to consider the domain of the original function as well. Any values of x that make the derivative zero or undefined but are not in the domain of the original function are not critical numbers. By carefully analyzing both the numerator and the denominator of the derivative, and considering the domain of the original function, we can accurately identify all the critical numbers, including the value of A in this case. This meticulous approach ensures that we do not miss any crucial points in our analysis.
With the critical number A in hand, we can now divide the real number line into intervals. These intervals are crucial because they allow us to analyze the function's behavior – whether it is increasing or decreasing – in each interval. The critical number acts as a dividing point, potentially marking a transition from increasing to decreasing or vice versa. In this specific case, we will have two intervals: (-∞, A) and (A, ∞). These intervals represent all real numbers less than A and all real numbers greater than A, respectively. To determine the function's behavior within each interval, we will employ a technique called the test value method. This involves selecting a test value within each interval and evaluating the derivative at that point. The sign of the derivative at the test value will indicate whether the function is increasing (positive derivative) or decreasing (negative derivative) in that interval. By systematically analyzing each interval, we can gain a comprehensive understanding of the function's overall behavior and identify its increasing and decreasing intervals.
Using Test Values to Determine Increasing and Decreasing Intervals
To ascertain whether the function f(x) is increasing or decreasing on the intervals (-∞, A) and (A, ∞), we employ the test value method. This method involves selecting a representative value, called a test value, within each interval and evaluating the derivative of the function at that point. The sign of the derivative at the test value provides valuable information about the function's behavior in that interval. If the derivative is positive at the test value, the function is increasing in that interval. Conversely, if the derivative is negative at the test value, the function is decreasing in that interval. The logic behind this method is that the sign of the derivative remains constant within an interval where the function is continuous and differentiable, and where the derivative does not cross zero or become undefined. By strategically choosing test values within each interval, we can efficiently determine the function's increasing and decreasing behavior without having to evaluate the derivative at every point. This method is a powerful tool in calculus for analyzing function behavior and understanding its graphical representation. In our case, we will choose a test value in (-∞, A) and another in (A, ∞), plug them into the derivative, and analyze the resulting signs to determine the intervals of increase and decrease.
Interpreting the Results
After evaluating the derivative at the test values in each interval, we carefully interpret the results to determine the increasing and decreasing behavior of the function f(x). If the derivative is positive in an interval, it signifies that the function is increasing in that interval. This means that as x increases, the value of f(x) also increases. On the graph of the function, this corresponds to the curve rising from left to right. Conversely, if the derivative is negative in an interval, it indicates that the function is decreasing in that interval. In this case, as x increases, the value of f(x) decreases. Graphically, this is represented by the curve falling from left to right. If the derivative is zero at a point, it suggests a potential local maximum, local minimum, or a horizontal inflection point. The critical number A, where the derivative is either zero or undefined, plays a crucial role in delineating the intervals of increasing and decreasing behavior. By analyzing the sign of the derivative in the intervals around A, we can determine whether A corresponds to a local maximum, a local minimum, or neither. This interpretation is essential for sketching the graph of the function and understanding its overall shape and behavior. The insights gained from this analysis are not only valuable for theoretical understanding but also have practical applications in optimization problems, where identifying intervals of increasing and decreasing behavior is crucial for finding maximum and minimum values.
In conclusion, finding the critical number A for the function f(x) = 6(x-4)^(2/3) is a vital step in understanding its behavior. By calculating the derivative, identifying where it is zero or undefined, and using test values in the resulting intervals, we can determine the intervals where the function is increasing and decreasing. This knowledge is crucial for sketching the graph of the function, finding local extrema, and solving optimization problems. The critical number A serves as a pivotal point, marking potential transitions in the function's behavior. The process of finding and interpreting critical numbers is a cornerstone of calculus and has broad applications in various fields. This exploration of f(x) = 6(x-4)^(2/3) serves as a valuable example of how calculus tools can be used to analyze and understand the behavior of functions, paving the way for deeper insights into mathematical concepts and their real-world applications. The systematic approach we've employed – calculating the derivative, finding critical points, and using test values – is a general method applicable to a wide range of functions, making this analysis a fundamental skill in calculus.
