Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the function, (b) use the graph to find the open intervals on which the function is increasing and decreasing, and (c) approximate any relative maximum or minimum values. $$f(x)=2 x^{2} e^{x+1}$$

Step-by-Step Solution

Answer

Plotting the function grants a visually graph to determine where the function is increasing or decreasing and to approximate relative extremas.

Step 1: Graph the Function

Insert the function \(f(x)=2x^{2}e^{x+1}\) into the graphing utility. Analyze the output graph.

Step 2: Find Open Intervals of Increase and Decrease

Analyze the graph and observe where the function increases and decreases. The function is increasing where the slope of the tangent line is positive, and decreasing where the slope of the tangent line is negative.

Step 3: Approximate Relative Maximum/Minimum

Look at the turning points of the graph, as these are where relative maxima and minima occur. For this, select the highest point for maxima and the lowest point for minima.

Key Concepts

Increasing and Decreasing Intervals

When analyzing a function, one of the vital aspects is understanding where the function is increasing or decreasing. An interval where a function increases means that the function values get larger as we move from left to right along the x-axis. Conversely, on a decreasing interval, these values become smaller.

To determine these intervals, we typically use the graph of a function. Here's how you can identify such intervals:

• Pay close attention to the slope of the tangent line at any given point on the function's graph. A positive slope indicates an increasing interval.
• A negative slope, on the other hand, points to a decreasing interval.
• If you're using a graphing utility, look for portions of the graph that rise (increasing) or fall (decreasing) when moving from left to right.

Utilizing these observations allows you to visually segment the function into increasing and decreasing intervals, which further aids in understanding the behavior of the function.

Relative Maximum and Minimum

At the heart of understanding functions are relative maximums and minimums, also referred to as local extrema. A relative maximum is a point where the function reaches a peak in its vicinity, meaning it is the highest point within some interval. Conversely, a relative minimum is the lowest point over a local interval.

To approximate these points on a graph:

• Look for the peaks of the 'hills' for relative maximums.
• At the base of the 'valleys', you'll find relative minimums.
• These points occur at turning points where the function changes direction.

By identifying these extrema, you can understand better the function's range of possible values and overall shape, which are crucial for applications in optimization problems.

Graphing Utilities

Graphing utilities are powerful tools for visualizing and analyzing functions. They help us to graph complex functions and to analyze their characteristics, like increasing/decreasing behavior and identifying relative maxima and minima. Here's what you need to know about them:

• Graphing utilities can handle a range of functions and provide a visual representation quickly.
• Beyond just plotting points, they include features to calculate derivatives, which aid in determining slopes for increasing and decreasing intervals.
• These tools let you zoom in on portions of the graph to see finer details and make approximations for relative extrema.

Using graphing utilities, you can make complex analyses far more intuitive and less time-consuming, providing an invaluable aid in both learning and practical applications.