Question

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$ f(x)=(x-1)^{2 / 3} $$

Short answer

The critical number of the function is \(x=1\). By evaluating the function at the critical numbers and endpoints, we can determine the absolute extrema. The results can be confirmed by using a graphing tool.

Step-by-step solution

Compute the derivative.

To identify the critical points, first take the derivative of the function \(f(x)=(x-1)^{2 / 3}\). Use the power rule, \(f'(x)=\frac{2}{3} (x-1)^{-1 / 3}\)

Find the critical numbers.

Next, find the values of \(x\) where \(f'(x) = 0\) or \(f'(x)\) does not exist. Here, \(f'(x)\) does not exist when \(x=1\). So, \(x=1\) is a critical number.

Determine the absolute extrema.

Evaluate the function at its critical number and at the endpoints of the closed interval. The largest and smallest of these values are the absolute maximum and minimum, respectively.

Verify the results graphically.

Use a graphing utility to plot the function in its domain. The absolute extrema of the function can be observed visually from the graph. The graph will touch its highest andlowest points at the absolute maximum and minimum points, respectively.

Discussion of absolute extrema.

The points at which the function reaches its highest and lowest points in the given interval are called the absolute extrema. These points provide important information about the behavior of the function.

Key concepts

Critical Numbers

When attempting to identify the absolute extrema of a function over a certain interval, a key step is to determine its critical numbers. Critical numbers occur at points where the derivative of the function either equals zero or is undefined. These are essential because they may correspond to local maxima, local minima, or points of inflection on the graph.

For instance, in the given exercise with function f(x)=(x-1)^{2/3}, critical numbers are crucial in locating potential extrema. You found that at x=1, the derivative of the function is undefined, making x=1 a critical number. Note that not all critical numbers will lead to an absolute extremum, but they are indispensable in the search for them. Furthermore, in functions involving roots or fractions, always verify where the derivative may be undefined, as in this example.

Derivative

The derivative of a function is a fundamental concept in calculus that indicates the rate of change of the function’s output with respect to its input. In other words, it shows how fast the function's value is changing at any given point, and it is pivotal for determining the function's increasing or decreasing behavior, as well as for locating critical numbers.

In your exercise, the derivative, denoted as f'(x) for f(x)=(x-1)^{2/3}, was computed using specific rules of differentiation. Through this process, you learned that the derivative helps in revealing the function's behavior, such as slopes of tangents, and in locating points where these slopes are zero or undefined, thus highlighting potential extremes in the function's graph.

Power Rule

A valuable tool for computing derivatives, especially when working with polynomial-like functions, is the power rule. The power rule simplifies the process of differentiation by stating that the derivative of x^n is nx^(n-1). This applies even when n is a fraction, which is often the case in functions with radicals or exponents that are not whole numbers.

In the exercise, you applied the power rule to differentiate (x-1)^(2/3), and found the derivative to be (2/3)*(x-1)^(-1/3). This step is fundamental to identifying critical numbers and understanding the function’s rate of change. For those new to calculus, practicing the power rule can greatly improve your speed and accuracy in finding derivatives.

Graphing Utility

Graphing utilities, such as graphing calculators or computer software, play a significant role in visualizing mathematical concepts and verifying analytical results. By plotting functions, you can observe their behavior such as continuity, extrema, and inflection points, which may not be immediately apparent through algebraic analysis alone.

In the context of the given exercise, after finding the critical numbers and potential extrema algebraically, a graphing utility was employed to provide a visual confirmation of the results. When you plot f(x)=(x-1)^{2/3}, you should see the absolute maximum and minimum of the function within the given interval, demonstrating the practical application of graphing utilities in comprehending the graphical aspects of calculus problems.