Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the function, approximate the relative minimum or maximum of the function, and estimate the open intervals on which the function is increasing or decreasing. \(f(x)=-x^{3}+3 x+1\)

Step-by-Step Solution

Answer

The relative maximum of the function \(f(x)= -x^3 + 3x + 1\) is around x=1.15, y=2.74, and the function doesn't have a relative minimum. The function increases on the interval (-∞, 1.15) and decreases on the interval (1.15, ∞). Please note these are approximations, exact values vary depending on the graphing technology used.

Step 1: Graphing the Function

Start by graphing the function \(f(x) = -x^{3} + 3x + 1\). A cubic function typically shapes like an 'S' or an 'inverted S' depending on whether the coefficient of \(x^3\) is positive or negative. In this case we expect an 'S' shaped graph because the coefficient of \(x^3\) is -1.

Step 2: Identify the Relative Maximum and Minimum

Once the function is graphed, we can visually inspect the graph to identify its high point (relative maximum) and low point (relative minimum). The highest and lowest points on the graph that do not continue endlessly to infinity are considered the relative maximum and minimum.

Step 3: Identify Intervals of Increase and Decrease

With the graph visible, we can simply 'read' the intervals where the function is increasing or decreasing. The function is increasing on an interval if as we move from left to right, the graph is moving upwards. Conversely, the function is decreasing on an interval if as we move from left to right, the graph is moving downwards.

Key Concepts

Relative Extrema

Understanding relative extrema is pivotal in graphing cubic functions. Relative extrema refer to the high and low points on a graph, which aren't necessarily the highest or lowest values of the function overall—these are known as absolute extrema. Instead, they are points where the function changes direction. To spot a relative maximum, look for a point where the graph transitions from increasing to decreasing; conversely, a relative minimum occurs where the graph transitions from decreasing to increasing.

In the context of our exercise with the function f(x) = -x^3 + 3x + 1, after using the graphing utility, we would observe the curve and look for these peaks and troughs. The importance of accurately identifying relative extrema lies in their utility for understanding the behavior of the function and in many practical applications, such as finding optimal solutions and predicting trends.

Intervals of Increase and Decrease

The intervals of increase and decrease in a function can be interpreted as where the function is either gaining or losing value, respectively. For cubic functions like ours, these intervals are crucial to comprehending the overall shape and direction of the graph.

A function is increasing on an interval if, as you move from left to right along the x-axis, the y-values rise. It is decreasing when the y-values fall over an interval as you move from left to right. To visualize this for f(x) = -x^3 + 3x + 1, we may see the graph rise from the left, reach a peak (relative maximum), then dip to form a valley (relative minimum), and finally drop off further to the right. Identifying these intervals allows students to anticipate the behavior of the function and apply this understanding to real-world scenarios.

Using Graphing Utilities

The advent of graphing utilities has made visualizing and working with cubic functions much more accessible. These tools are software or online platforms that enable us to plot functions like f(x) = -x^3 + 3x + 1 with precision and ease. Upon inputting the function, the utility generates a graph, which can then be used to approximate relative extrema and determine intervals of increase or decrease.

For students, it's important to familiarize themselves with the functionalities of graphing utilities—not only for plotting but also for evaluating, zooming, and finding specific points. Being proficient with these tools can make detecting patterns and characteristics in the function's behavior significantly easier.

Cubic Function Characteristics

Cubic functions exhibit a distinctive set of characteristics due to their form f(x) = ax^3 + bx^2 + cx + d. Chief among these characteristics is their shape: they tend to have an 'S' or an 'inverted S' curve, known as the graph's inflection point, where it changes concavity. Furthermore, they have either one relative maximum and one relative minimum or none at all; in the case of f(x) = -x^3 + 3x + 1, we expect an 'S' shape because the leading coefficient is negative.

Moreover, cubic functions are always continuous and smooth—they do not have sharp corners or breaks. Lastly, the end behavior of cubic functions is such that as x approaches positive or negative infinity, f(x) will also shoot off to infinity, depending on the sign of the leading coefficient. This knowledge aids in sketching a rough graph of the function before using graphing utilities for a more precise depiction.