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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 4.10
Chapter 4, Problem 4.10

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.


y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)

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First, identify the natural domain of the function y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2). The denominator must not be zero, so solve 𝓍² + 2𝓍 + 2 = 0 to find any restrictions on 𝓍.
Next, find the derivative of the function y with respect to 𝓍 to determine critical points. Use the quotient rule: if y = u/v, then y' = (u'v - uv') / vΒ², where u = 𝓍 + 1 and v = 𝓍² + 2𝓍 + 2.
Set the derivative equal to zero to find critical points. Solve the equation (u'v - uv') = 0 for 𝓍, which will give you the values of 𝓍 where the slope of the tangent is zero, indicating potential extreme values.
Evaluate the second derivative to determine the concavity at the critical points. If the second derivative is positive at a critical point, the function has a local minimum there; if negative, a local maximum.
Finally, analyze the behavior of the function as 𝓍 approaches the boundaries of the domain or infinity to determine if there are any absolute extreme values. Consider the limits of y as 𝓍 approaches any boundary values or infinity.

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Key Concepts

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

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential candidates for local extrema. To find them, differentiate the function with respect to x, set the derivative equal to zero, and solve for x. Additionally, check where the derivative does not exist, as these points may also indicate critical points.
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Critical Points

First Derivative Test

The First Derivative Test helps determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function. If the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it's a local minimum.
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The First Derivative Test: Finding Local Extrema

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, like the one given, the domain excludes values that make the denominator zero. Understanding the domain is crucial for identifying where to look for extrema, as it defines the interval over which the function is analyzed.
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Finding the Domain and Range of a Graph
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.

39. y = 8 / (xΒ² + 4) (Witch of Agnesi)

Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(3tΒ² + t/2) dt

Textbook Question

The intensity of illumination at any point from a light source is proportional to the square of the reciprocal of the distance between the point and the light source. Two lights, one having an intensity eight times that of the other, are 6 m apart. How far from the stronger light is the total illumination least?

Textbook Question

Theory and Examples


Determine the values of constants a and b so that f(x) = axΒ² + bx has an absolute maximum at the point (1,2).

Textbook Question

30. Find a positive number for which the sum of its reciprocal and four times its square is the smallest possible.

Textbook Question

Theory and Examples

67. An inequality for positive integers Show that if a, b, c, and d are positive integers, then

[(a^2+1)(b^2+1)(c^2+1)(d^2+1)]/abcd β‰₯ 16