Problem 52 - Precalculus with Limits

Question

In Exercises 47-56, (a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part (a). \(f(x) = 3x^4 - 6x^2\)

Step-by-Step Solution

Answer

The function \(f(x) = 3x^4 - 6x^2\) is decreasing from \(-\infty\) to \(0\), and increasing from \(0\) to \(+\infty\).

Step 1: Graph the function

Use a graphical representation to plot the function \(f(x) = 3x^4 - 6x^2\). The graph will provide a visual illustration of the function's behavior which can be used to identify intervals where the function is increasing, decreasing, or constant.

Step 2: Determine the intervals

Identify the intervals where the function is increasing, decreasing, or constant. From the graph, the function is decreasing from \(-\infty\) to \(0\) and from \(0\) to \(+\infty\), it is increasing.

Step 3: Verify with a table

Make a table of function values for the interval to verify if the function is increasing or decreasing as indicated by the graph. Choose values for \(x\) within the intervals identified, calculate \(f(x)\), and notice how \(f(x)\) changes as \(x\) increases.

Key Concepts

Understanding Graphing Utilities

Graphing utilities are essential tools for visualizing mathematical functions, allowing students to see the behavior of equations like f(x) = 3x^4 - 6x^2 in a graphical format.

Imagine plotting points on a Cartesian plane—graphing utilities automate this process, providing a smooth curve that represents all possible points (x, f(x)) for the function. Some popular graphing utilities include software like Desmos, GeoGebra, and even features found in graphing calculators.

Why use them? Because they not only help us understand the shape and trajectory of functions but they also make it easier to identify key characteristics such as maximum and minimum points, symmetry, and, importantly for our exercise, intervals of increase and decrease.

Intervals of Increase and Decrease

Identifying intervals of increase and decrease is a foundational skill in understanding function behavior. For a function like f(x) = 3x^4 - 6x^2, these intervals indicate where the output of the function, f(x), grows or diminishes as x moves along the number line.

An increasing interval means as x gets larger, so does f(x), while a decreasing interval implies f(x) gets smaller as x increases. In a graph, this can be seen as the portion of the curve going upwards for an increasing function and downwards for a decreasing function.

Identifying these intervals using a graphing utility can initially be done visually, but it is good practice to verify this by calculating the function's values at certain points—a task where a table of values becomes invaluable.

Creating a Table of Values

A table of values is a systematic way to validate the behavior of functions. After using a graphing utility, you would create a table that lists several ordered pairs (x, f(x)).

For the given function f(x) = 3x^4 - 6x^2, select x-values within the identified intervals of increase and decrease. Calculate and tabulate the respective f(x) for each x. By examining how f(x) changes as x progresses, you can confirm or refute your initial visual assessment of the function's behavior.

Remember, consistency is crucial. If for increasing x-values, f(x) consistently gets larger, the function is increasing, and vice versa for a decreasing function. Regular use of tables of values reinforces understanding and leads to a more robust grasp of mathematical concepts.