Book cover for Calculus of a Single Variable

Calculus of a Single Variable

Ron Larson, Bruce Edwards

ISBN #9781285060286

10th Edition

6,214 Questions

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101,749 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers how to find the extreme values (maximum and minimum) of functions on an interval using derivatives. Students learn that absolute extrema can occur at endpoints or critical points, and that relative extrema occur only at critical numbers. The Extreme Value Theorem guarantees extrema for continuous functions on closed intervals. Additionally, Rolle’s Theorem provides conditions under which a function has a horizontal tangent, highlighting the importance of differentiability. Understanding these concepts is key for solving optimization problems in both theoretical and practical contexts.

Learning Objectives

1

Describe the definitions of absolute and relative extrema on an interval and explain how endpoints and critical numbers play a role.

2

Demonstrate how to apply the Extreme Value Theorem to find extrema on a closed interval.

3

Identify and compute critical numbers and use them to determine relative extrema of functions.

4

Explain Rolle’s Theorem, including its conditions and implications for differentiable functions on closed intervals.

5

Apply differentiation techniques to solve optimization problems in real-world applications.

Key Concepts

CONCEPT

DEFINITION

Absolute (Global) Extremum

The absolute maximum (or minimum) of a function on an interval is the highest (or lowest) value that the function attains on that interval. These can occur at interior points or endpoints.

Relative (Local) Extremum

A relative maximum or minimum is a value in an open interval such that there exists a neighborhood around that point where the function values are all less (or greater) than or equal to the function value at that point.

Critical Number

A critical number of a function f is a number c in the domain of f where either f'(c) = 0 or f'(c) does not exist. Critical numbers are potential candidates for relative extrema.

Extreme Value Theorem

A continuous function on a closed interval [a, b] must attain both an absolute minimum and an absolute maximum value somewhere on that interval.

Rolle’s Theorem

If a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that f'(c) = 0.

Example Problems

Example 1

Finding the Value of the Derivative at Relative Extrema In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum. $$ f(x)=\frac{x^{2}}{x^{2}+4} $$

Example 2

Finding the Value of the Derivative at Relative Extrema In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum. $$ f(x)=\cos \frac{\pi x}{2} $$

Example 3

Finding the Value of the Derivative at Relative Extrema In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum. $$ g(x)=x+\frac{4}{x^{2}} $$

Example 4

Finding the Value of the Derivative at Relative Extrema In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum. $$ f(x)=-3 x \sqrt{x+1} $$

Example 5

Finding the Value of the Derivative at Relative Extrema In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum. $$ f(x)=(x+2)^{2 / 3} $$
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Step-by-Step Explanations

QUESTION

How do you find the absolute maximum and minimum of a function f(x) on the closed interval [a, b]?

STEP-BY-STEP ANSWER:

Step 1: Differentiate the function f(x) to obtain f'(x).
Step 2: Identify the critical numbers by solving f'(x) = 0 and noting points where f'(x) does not exist, ensuring the critical numbers are in the interval [a, b].
Step 3: Evaluate the function f(x) at each critical number as well as at the endpoints a and b.
Step 4: Compare all function values; the smallest value is the absolute minimum and the largest value is the absolute maximum.
Final Answer: The extrema are the lowest and highest values from the evaluations.

Finding Extrema on a Closed Interval

QUESTION

Given a function f(x) that is continuous on [a, b] and differentiable on (a, b) with f(a) = f(b), how do you verify Rolle's Theorem?

STEP-BY-STEP ANSWER:

Step 1: Confirm that f(x) is continuous on the interval [a, b].
Step 2: Confirm that f(x) is differentiable on the open interval (a, b).
Step 3: Verify that f(a) equals f(b).
Step 4: Conclude that there exists at least one c in (a, b) such that f'(c) = 0, as guaranteed by Rolle’s Theorem.
Final Answer: The existence of a point c with f'(c) = 0 validates the theorem under the given conditions.

Applying Rolle's Theorem

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Common Mistakes

  • Failing to evaluate the function at the endpoints when finding extrema on a closed interval.
  • Assuming that every critical number corresponds to a relative extremum, even when the derivative does not change sign.
  • Confusing absolute extrema with relative extrema, particularly in open intervals where endpoints are not included.
  • Overlooking points where the derivative does not exist, which may still be critical numbers.
  • Misapplying Rolle’s Theorem by not verifying all its conditions (continuity on [a, b], differentiability on (a, b) and equal endpoints).