Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

The First Derivative Test

Calculus

5. Graphical Applications of Derivatives

The First Derivative Test

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2:39 minutes

Problem 4.1.54

Textbook Question

Theory and Examples

In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.

y = 3x + tan x

Verified step by step guidance

Identify the natural domain of the function y = 3x + tan(x). The function tan(x) is undefined at odd multiples of π/2, so the natural domain is all real numbers except x = (2n+1)π/2, where n is an integer.

Consider the behavior of the function as x approaches the points where tan(x) is undefined. As x approaches (2n+1)π/2 from the left, tan(x) approaches negative infinity, and from the right, tan(x) approaches positive infinity. This indicates that the function y = 3x + tan(x) has vertical asymptotes at these points.

Analyze the behavior of the function as x approaches positive and negative infinity. As x approaches positive or negative infinity, the linear term 3x dominates, causing the function to increase or decrease without bound.

Since the function has vertical asymptotes and increases or decreases without bound as x approaches infinity or negative infinity, it does not attain a highest or lowest value on its natural domain.

Conclude that the function y = 3x + tan(x) has neither an absolute minimum nor an absolute maximum on its natural domain due to the presence of vertical asymptotes and unbounded behavior at infinity.

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Key Concepts

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

Natural Domain

The natural domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function y = 3x + tan x, the natural domain excludes values where tan x is undefined, such as x = π/2 + kπ, where k is an integer. Understanding the domain is crucial for analyzing the behavior of the function across its entire range.

Absolute Minimum and Maximum

An absolute minimum or maximum refers to the lowest or highest value a function can attain on a given interval or domain. For y = 3x + tan x, determining the existence of absolute extrema involves analyzing the function's behavior and limits, especially considering the periodic nature and asymptotes of the tangent function, which can lead to unbounded behavior.

Behavior of Trigonometric Functions

Trigonometric functions like tan x have unique properties, such as periodicity and asymptotes, which affect their graphs and values. The tangent function has vertical asymptotes at odd multiples of π/2, causing it to approach infinity. This behavior is key to understanding why y = 3x + tan x may not have absolute extrema, as the function can increase or decrease without bound near these asymptotes.