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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.3.18
Capitolo 4, Problema 4.3.18

Identifying Extrema


In Exercises 15–18:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local and absolute extreme values, if any, saying where they occur.


Graph of a function showing local minima and maxima, with labeled axes and a curve illustrating increasing and decreasing intervals.

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To determine where the function is increasing or decreasing, observe the graph of the function. The function is increasing where the slope of the graph is positive and decreasing where the slope is negative.
Identify the intervals where the function is increasing. From the graph, the function appears to be increasing on the interval (-2, 1).
Identify the intervals where the function is decreasing. From the graph, the function appears to be decreasing on the intervals (-∞, -2) and (1, ∞).
To find local extrema, look for points where the function changes from increasing to decreasing or vice versa. The graph shows a local maximum at x = -2 and a local minimum at x = 1.
Check for absolute extrema by comparing the values of the function at critical points and endpoints. The graph suggests an absolute maximum at x = -2 and an absolute minimum at x = 1.

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Increasing and Decreasing Intervals

A function is increasing on an interval if, for any two numbers x1 and x2 within that interval, x1 < x2 implies f(x1) < f(x2). Conversely, it is decreasing if x1 < x2 implies f(x1) > f(x2). Identifying these intervals involves analyzing the function's derivative: where it is positive, the function is increasing, and where it is negative, the function is decreasing.
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Determining Where a Function is Increasing & Decreasing

Local and Absolute Extrema

Local extrema are points where a function reaches a minimum or maximum value within a neighborhood, while absolute extrema are the highest or lowest points over the entire domain. To find these, examine critical points where the derivative is zero or undefined, and evaluate the function's behavior at these points and endpoints of the domain.
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Finding Extrema Graphically

Critical Points

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. To determine the nature of these points, use the first or second derivative test. The first derivative test involves checking sign changes around the critical points, while the second derivative test uses concavity to identify extrema.
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Critical Points
Pratica correlata
Domanda del libro di testo

In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

y = 1 / (x² - 1)

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Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = x − 6√(x − 1)

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Domanda del libro di testo

26. Constructing cylinders Compare the answers to the following two construction problems.

a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?

b. The same sheet is to be revolved about one of the sides of length y to sweep out the cylinder as shown in part (b) of the figure. What values of x and y give the largest volume?

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Domanda del libro di testo

107. Marginal cost The accompanying graph shows the hypothetical cost c=f(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?

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Checking the Mean Value Theorem


Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.


f(x) =√(x − 1), [1, 3]

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Domanda del libro di testo

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

7. y=sin|x|, -2π≤x≤2π

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