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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.R.5c
Capitolo 4, Problema 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.

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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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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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The 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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The Second Derivative Test: Finding Local Extrema
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