Problem 26 - College Algebra and Calculus: An Applied Approach

Question

In Exercise, use a graphing utility to estimate graphically all relative extrema of the function. $$ f(x)=3 x^{3}+5 x^{2}-2 $$

Step-by-Step Solution

Answer

The relative extrema can be found by inspecting the graph of the function \(f(x) = 3x^{3} + 5x^{2} - 2\). These extrema occur at the points where the graph has local maximums and minimums. The exact points need a graphing utility to be determined.

Step 1: Understanding Relative Extrema

Relative extrema of a function are the local maximum and minimum points in its graph, i.e., points where the function is higher/lower than all nearby points. We will find these points graphically using a graphing utility.

Step 2: Graph the Function

Graph the function \( f(x) = 3x^{3} + 5x^{2} - 2 \) using a graphing utility. Remember, the x-axis represents the input and the y-axis represents the output.

Step 3: Identify the Relative Extrema

Examine the graph to identify the highest and lowest points within each local section. These will be the relative extrema. The relative maximum is the highest point within a local section and the relative minimum is the lowest point within a local section.

Key Concepts

Graphing Utility

A graphing utility is a tool used to visualize mathematical functions. This can be a physical calculator, software on a computer, or an online tool. Using a graphing utility helps you see a graphical representation of a function, which is essential for understanding its behavior. When you plug the function into a graphing utility:

• Input the mathematical expression exactly as it's given.
• Adjust the viewing window to ensure you clearly see the important parts of the graph.
• Use the zoom functions to get a closer look at areas of interest.

Having the graph in front of you allows you to spot features like intercepts, slopes, curvature, and, most importantly for this exercise, the relative extrema. Graphs provide a visual way to estimate these key points, which can then be confirmed with further calculations if needed.

Local Maximum

A local maximum is a point on a graph where the function reaches a peak within a certain neighborhood. It doesn’t have to be the highest point overall, but it is the highest within that small section of the graph. To identify a local maximum:

• Look for a peak or crest on the graph.
• Check that for points on either side of it, the function is decreasing.
• This point should look like the top point of a hill.

In our exercise, we use a graphing utility to identify these peaks. Notice that local maxima aren't necessarily the highest points overall on the graph. Instead, they're simply the highest where you zoom in, providing crucial insight into the function's structure and behavior. This concept helps in understanding turning points, decision-making areas in calculus, and even real-world data modeling.

Local Minimum

Conversely, a local minimum is a point where the function reaches its lowest value within a specific area on the graph. Similar to local maxima, these points don't have to be the lowest overall, just lower than the surrounding points within their neighborhood. To spot a local minimum, follow these steps:

• Identify a valley or trough on the graph.
• Ensure that the function is increasing on both sides of this point.
• This point looks like the bottom of a shallow bowl.

By using the graphing utility in our exercise, you can visually find these low points with relative ease. Locating local minima is important in various applications, from optimization problems in calculus to data trends and economics, as they represent critical points where behavior shifts or where calculations might need adjustments.