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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 60
Chapter 4, Problem 60

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
60. y = 5 / (x⁴ + 5)

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Step 1: Analyze the function y = 5 / (x⁴ + 5). Start by identifying the domain of the function. Since the denominator x⁴ + 5 is always positive and never zero, the function is defined for all real numbers (-∞, ∞).
Step 2: Compute the first derivative y' to find critical points and identify local extreme points. Use the quotient rule: \( y' = \frac{(x^4 + 5)(0) - 5(4x^3)}{(x^4 + 5)^2} = \frac{-20x^3}{(x^4 + 5)^2} \). Set y' = 0 to find critical points, which occurs when the numerator -20x³ = 0. Solve for x.
Step 3: Compute the second derivative y'' to determine concavity and locate inflection points. Differentiate y' using the quotient rule again. Simplify the expression for y'' and set it equal to 0 to find where the concavity changes. Solve for x to find potential inflection points.
Step 4: Evaluate the function y at the critical points and endpoints (if applicable) to determine the absolute extreme points. Since the domain is all real numbers, there are no finite endpoints, but check the behavior of y as x approaches ±∞ to determine if absolute extrema exist.
Step 5: Use the information from the first and second derivatives to sketch the graph of the function. Mark the critical points, inflection points, and any absolute extreme points on the graph. Note the behavior of the function as x approaches ±∞ and the symmetry of the function (if any).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the behavior of the function. This includes identifying key features such as intercepts, local extrema (maximum and minimum points), and inflection points where the concavity changes. Understanding how to analyze the function's equation helps in determining these characteristics effectively.
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Graph of Sine and Cosine Function

Local and Absolute Extrema

Local extrema refer to points in the domain of a function where it reaches a local maximum or minimum compared to nearby points. Absolute extrema, on the other hand, are the highest or lowest points over the entire domain of the function. Finding these points typically involves using the first derivative test to identify critical points and evaluating the function at these points.
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Finding Extrema Graphically

Inflection Points

Inflection points are points on the graph of a function where the curvature changes, indicating a shift in concavity from concave up to concave down or vice versa. To find inflection points, one must analyze the second derivative of the function. If the second derivative changes sign at a point, that point is classified as an inflection point, which is crucial for understanding the overall shape of the graph.
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Critical Points
Related Practice
Textbook Question

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

56. y = x² + 2/x

Textbook Question

Sketch the graphs of the rational functions in Exercises 53–60.

y = (x2 + 1) / x                              

Textbook Question

A customer has asked you to design an open-top rectangular stainless steel vat. It is to have a square base and a volume of 32 ft³ , to be welded from quarter-inch plate, and to weigh no more than necessary. What dimensions do you recommend?

Textbook Question

An isosceles triangle has its vertex at the origin and its base parallel to the x-axis with the vertices above the axis on the curve y = 27 - x2. Find the largest area the triangle can have.

Textbook Question

Sketch the graphs of the rational functions in Exercises 53–60.

y= (x + 1) / (x - 3)

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Textbook Question

The sum of two nonnegative numbers is 36. Find the numbers if

a. the difference of their square roots is to be as large as possible.

b. the sum of their square roots is to be as large as possible.