Problem 63 - 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. $$f(x)=x^{3} e^{x}$$

Step-by-Step Solution

Answer

Use a graphing tool to plot the function, then observe the graph to find out the intervals where the function is increasing or decreasing, and finally, approximate any relative maximum or minimum points. These steps help in understanding the nature and behaviour of the function.

Step 1: Graphing the Function

Use a graphing utility to plot the function \(f(x)=x^{3} e^{x}\). Evaluate a few data points to get an understanding about the relation.

Step 2: Determining Intervals of Increase and Decrease

Observe the graph to find intervals where the function is increasing or decreasing. A function is said to be increasing on an interval if, for any two numbers in the interval, when the first number is less than the second number, the function value of the first number is less than that of the second number. Similarly, a function is decreasing on an interval if, for any two numbers in that interval, when the first number is less than the second one, the function value of the first number is more than that of the second number.

Step 3: Approximating Relative Maximum or Minimum Values

Look at the graph to approximate any relative maximum or minimum points. A relative maximum is the highest point in a particular section of the graph. Similarly, a relative minimum is the lowest point in a particular section of the graph. If the function changes from increasing to decreasing at a point, that point is a relative maximum. If the function changes from decreasing to increasing at a point, that function is a relative minimum.

Key Concepts

Increasing and Decreasing Intervals

Understanding when a function is increasing or decreasing is essential in analyzing its behavior. A function is said to be **increasing** on an interval if, for every pair of numbers within that interval, the function's output is larger for the larger input. In simpler terms, as you move from left to right along the interval, the graph of the function kind of goes uphill. Conversely, a function is **decreasing** on an interval if, within that interval, the larger input yields a smaller output compared to a smaller input. You can imagine the graph going downhill as you move from left to right. Key Points:

• Identify the intervals by observing the graph's direction.
• Evaluate specific data points if needed.
• Any switch from increasing to decreasing or vice versa often hints at special points like maxima or minima.

Examining these intervals gives us insights into the function's behavior, helping to predict or understand real-world phenomena it may represent.

Relative Maximum and Minimum

Relative maxima and minima provide critical insights into the function's performance at certain points. A **relative maximum** occurs at a point where the function shifts from increasing to decreasing. Visually, this looks like the peak or top of a hill in the graph. A **relative minimum** happens at a point where the function moves from decreasing to increasing, resembling a valley bottom in the graph. These points are "relative" because they don't necessarily have to be the highest or lowest points on the entire graph but are local to a certain interval. **Identifying Relative Extrema**

• Use the graph to spot where changes in direction occur.
• Approximate the precise values unless exact calculations are needed.
• These points can be crucial for understanding thresholds or turning points in practical scenarios.

Recognizing and interpreting these points can provide insights into optimal conditions or crucial shifts in patterns modeled by the function.

Graphing Utilities

Using graphing utilities is a practical way to visualize complex functions like the one given, \( f(x) = x^3 e^x \). These tools allow you to easily plot functions and gain a visual understanding of their behavior with minimal manual calculation.With graphing utilities, you can:

• Plot graphs swiftly and precisely.
• Adjust viewing windows to focus on areas of interest.
• Zoom in on specific intervals to see details more clearly.

**Benefits of Using Graphing Utilities:**

• Instantly identify increasing and decreasing intervals.
• Quickly approximate relative maxima and minima.
• Easily handle functions that would be tedious to plot by hand.

Graphing utilities simplify complex analyses, making them accessible and manageable for users at all levels, from students learning the basics to professionals conducting high-level research.