Textbook Question
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
e. On what intervals (approximately) is f concave up?
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.R.5c
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
c. Determine where f has local maxima and minima.
Verified step by step guidance1
Identify the critical points of the function f by finding where the derivative f' is equal to zero or undefined. These points are potential locations for local maxima or minima.
Examine the sign of the derivative f' around each critical point. If f' changes from positive to negative at a critical point, f has a local maximum there. If f' changes from negative to positive, f has a local minimum.
Use the second derivative f'' to confirm the nature of each critical point. If f'' is positive at a critical point, f has a local minimum there. If f'' is negative, f has a local maximum.
Check the endpoints of the domain if the function is defined on a closed interval, as local extrema can also occur at these points.
Summarize the locations of all local maxima and minima based on the analysis of f' and f''.
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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 occur where the derivative of a function, ƒ', is either zero or undefined. These points are essential for identifying local maxima and minima, as they represent potential locations where the function's slope changes. To find these points, one typically sets ƒ' equal to zero and solves for the variable.
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Critical Points
First Derivative Test
The First Derivative Test is a method used to determine whether a critical point is a local maximum, local minimum, or neither. By analyzing the sign of the derivative before and after the critical point, one can conclude that if ƒ' changes from positive to negative, a local maximum occurs, while a change from negative to positive indicates a local minimum.
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07:09The First Derivative Test: Finding Local Extrema
Second Derivative Test
The Second Derivative Test provides another way to classify critical points by examining the concavity of the function. If the second derivative, ƒ'', is positive at a critical point, the function is concave up, indicating a local minimum. Conversely, if ƒ'' is negative, the function is concave down, suggesting a local maximum.
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06:02The Second Derivative Test: Finding Local Extrema
Related Practice
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