Problem 14 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a graphing utility to graph the function. Then find all relative extrema of the function. $$ f(t)=(t-1)^{1 / 3} $$

Step-by-Step Solution

Answer

The function \(f(t)=(t-1)^{1 / 3}\) has no relative extrema.

Step 1: Plot Graph Using Graphing Utility

A graph of the function \(f(t)=(t-1)^{1 / 3}\) can be plotted using a graphing utility like Desmos or GeoGebra. When you input the equation into the graphing utility, you should get a curve showing the shape of the function.

Step 2: Identify Possible Extrema

Analyze the graph to identify any minimum or maximum points. Relative extrema occur at the highest or lowest points of the curves. For the function \(f(t)=(t-1)^{1 / 3}\), from the graph, it can be seen there is no local maximum or minimum point since the function is monotonically increasing.

Step 3: Find the Value of Extrema

Since there is no local maximum or minimum points for the function \(f(t)=(t-1)^{1 / 3}\), then there are no relative extrema.

Key Concepts

Relative Extrema

Relative extrema refer to the points on a graph where a function reaches a local maximum or minimum value. These are the "hills" and "valleys" that may occur throughout the domain of a function.
When observing a graph:

• A local maximum is the highest point in its immediate neighborhood.
• A local minimum is the lowest point in its immediate vicinity.

Relative extrema are crucial in understanding the behavior of functions over an interval. They help us determine where the function increases or decreases. However, it is important to note that not all functions possess a relative extrema. For instance, functions that are monotonic (always increasing or always decreasing) may not have any local maxima or minima, as they continue in one direction without any fluctuations in height.

Plotting Graphs

Plotting graphs is a fundamental skill in analyzing and understanding the behavior of functions. It provides a visual representation of how the function behaves across different values of its variable.
Using graphing utilities such as Desmos or GeoGebra makes this task easier as they allow you to visualize the function effortlessly:

• Input the function equation into the graphing tool.
• Observe the shape formed by the plotted graph.
• Identify key features such as intercepts, turning points, and behavior at infinity.

Plotting is not only helpful for visual confirmation but also essential for identifying relative extrema and understanding the overall behavior of the function. Through graphical analysis, students can intuitively grasp complex concepts that equations alone may not clearly convey.

Monotonic Functions

A monotonic function is one that is entirely non-decreasing or non-increasing over its domain. If a function never decreases, it is said to be **monotonically increasing**, and if it never increases, it is called **monotonically decreasing**.
This concept is exemplified in the function \(f(t)=(t-1)^{1 / 3}\), which exhibits a distinct behavior.

• The function is continuously increasing as "t" increases.
• Its derivative is either positive or zero, indicating a lack of decrease.

Monotonic functions are significant since they do not have any relative extrema. This makes them predictable in terms of their direction and can be an advantageous property when solving real-world problems where a continuous increase or decrease is expected.