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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.5a
Capitolo 4, Problema 4.R.5a

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
a. Find the critical points of f and determine where f is increasing and where it is decreasing.

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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 are the x-values where the graph of f' crosses the x-axis or has discontinuities.
Determine the intervals where the function f is increasing by analyzing the sign of f'. If f' is positive on an interval, then f is increasing on that interval.
Determine the intervals where the function f is decreasing by analyzing the sign of f'. If f' is negative on an interval, then f is decreasing on that interval.
Use the second derivative f" to confirm the nature of the critical points. If f" is positive at a critical point, it indicates a local minimum, and if f" is negative, it indicates a local maximum.
Summarize the findings by listing the critical points and specifying the intervals of increase and decrease for the function f, based on the analysis of f' and f".

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima and minima, as they indicate where the function's slope changes. To find critical points, one typically sets the first derivative, f', equal to zero and solves for the variable.
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Critical Points

Increasing and Decreasing Intervals

A function is increasing on an interval where its derivative, f', is positive, and decreasing where f' is negative. By analyzing the sign of the derivative around the critical points, one can determine the intervals of increase and decrease. This information is crucial for understanding the overall behavior of the function.
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Determining Where a Function is Increasing & Decreasing

Second Derivative Test

The second derivative test involves using the second derivative, f'', to determine the concavity of the function at critical points. If f'' is positive at a critical point, the function is concave up, indicating a local minimum; if f'' is negative, the function is concave down, indicating a local maximum. This test helps confirm the nature of critical points found using the first derivative.
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The Second Derivative Test: Finding Local Extrema
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