Problem 59 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use a graphing calculator or a CAS to plot the graph of each of the following functions on \([-1,7]\). Determine the coordinates of any global extrema and any inflection points. You should be able to give answers that are accurate to at least one decimal place. (a) \(f(x)=x \sqrt{x^{2}-6 x+40}\) (b) \(f(x)=\sqrt{|x|}\left(x^{2}-6 x+40\right)\) (c) \(f(x)=\sqrt{x^{2}-6 x+40} /(x-2)\) (d) \(f(x)=\sin \left[\left(x^{2}-6 x+40\right) / 6\right]\)

Step-by-Step Solution

Answer

Plot each function, then check the graph for extrema and inflection points.

Step 1: Understand the Functions

First, clarify what each function looks like, and identify their domains. For function (a), check that it's defined for all x in - [-1,7]. Functions (b) and (c) are defined similarly since - they also depend on the expression under the square root. - For function (b), note the modulus sign in the square - root term. For (c), ensure to account for division by zero. Function (d) can be plotted directly since the sin function is defined for all real numbers.

Step 2: Plot Function (a)

Use a graphing calculator or CAS to plot the graph of - function (a): \[f(x) = x \sqrt{x^2 - 6x + 40}\]\- on the interval [-1, 7]. Check for any peaks (extrema) and - inflection points by analyzing where the function changes - behavior.

Step 3: Identify Extrema and Inflection for (a)

By observing the graph, determine any global maxima or minima - within the domain. Also, identify changes in concavity as - inflection points. Record these as coordinates.

Step 4: Plot Function (b)

Next, graph \[f(x) = \sqrt{|x|}\left(x^2 - 6x + 40\right)\]\- within the same interval. Note where peaks and - changes in concavity occur as before.

Step 5: Identify Extrema and Inflection for (b)

From the plot, determine the global extrema and - points where the curve chooses in concavity for - function (b). Capture these points.

Step 6: Plot Function (c)

Using the graphing tool, plot \[f(x) = \frac{\sqrt{x^2 - 6x + 40}}{x-2}\] - - Ensure to note any undefined points or disruptions in the graph, - particularly around x=2.

Step 7: Identify Extrema and Inflection for (c)

Check the graph to detect any maxima, minima, and changes - in curvature for function (c), excluding any discontinuities.

Step 8: Plot Function (d)

Graph \[f(x) = \sin\left(\frac{x^2 - 6x + 40}{6}\right)\]\- over the specified domain. Identify where the sine function peaks - and where concavity shifts on this curve.

Step 9: Identify Extrema and Inflection for (d)

Using the plot, note where the highest and lowest points occur - for the sine function within the interval, and any inflection - points.

Key Concepts

Graphing Calculators

Graphing calculators and Computer Algebra Systems (CAS) are remarkable tools for students of calculus. They allow you to visualize functions easily, saving time on manual plotting and calculations. To use them effectively for analyzing functions, especially complicated ones, follow these steps:

• Input the function accurately, ensuring all parentheses and operators are correct. For example, for function (a), ensure you enter it as \(f(x) = x \sqrt{x^2 - 6x + 40}\).


• Set the domain correctly, which in this case is \([-1,7]\). Most graphing calculators allow you to adjust the viewing window to focus only on necessary portions of the graph.


• Use the tools provided by the calculator to find extrema and inflection points. These can include maximum and minimum finding functions or slope calculations to identify changes in concavity.

These devices enable you to easily manage the graphical analysis of functions, which is essential for visual learning and understanding complex behaviors in calculus.

Global Extrema

In calculus, global extrema are the highest and lowest values a function can reach over a certain interval. These are key points on a graph and they help in understanding the overall behavior of a function:

• Global maximum: The highest point on a graph over a given interval. At this point, the function's value is greater than at any other point in that interval.


• Global minimum: The lowest point on a graph over the interval, where the function's value is the smallest.

To find global extrema:

• Look at the graph carefully, noting peaks and troughs.


• Utilize your graphing calculator’s features to get precise coordinates.


• Ensure that the extrema are indeed "global" by checking if the function values return to lower or higher elsewhere in the given range.

Notably, function (c) from the exercise may have undefined points that can affect global extrema, particularly since the denominator \(x-2\) can be zero, leading to discontinuities that need consideration.

Inflection Points

Inflection points occur where a function changes its concavity, either from concave up to concave down or vice versa. This is crucial because these points often indicate significant changes in the behavior of a function:

• To identify them, look for points where the second derivative equals zero or is undefined, and test the sign change of the second derivative around these points.


• On a graph, these are spots where the curve appears to shift direction, giving a clue about where this change occurs in the curvature of the function.

Using a graphing calculator, finding inflection points involves visual inspection for curve shifts and sometimes calculus techniques for specifying changes in concavity numerically. Make sure to check these points with exactitude using features that might allow for derivative calculations on your graphing calculator. For functions in the exercise, particularly (b) and (d), these shifts in curvature will be quite prominent.

Function Analysis

Analyzing a function comprehensively requires a combination of techniques that include finding limits, derivatives, and examining graphs:

• Domain and Range: Establish the domain and range to know where the function is defined and what values it can take.


• Continuity: Determine if the function is continuous over its domain, particularly around critical points.


• Derivatives: Use the first and second derivatives to find and verify critical points, assists in solving for maxima, minima, and inflection points.

Function analysis is essential for understanding the nature of each part of the function and how different components impact it. For example, function (c) containing division by (x-2) highlights the need to scrutinize for removable discontinuities or asymptotic behavior. This approach allows deeper insights into the function's whole behavior within a defined interval.