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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.1.11
Chapter 4, Problem 4.1.11

Finding Extrema from Graphs


In Exercises 11–14, match the table with a graph.



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Identify the critical points from the table where the derivative f'(x) is zero. These points are x = a and x = b.
Since f'(a) = 0 and f'(b) = 0, these points could be local maxima, minima, or saddle points. We need to analyze the graph to determine the nature of these points.
For x = c, f'(c) = 5, which indicates that the function is increasing at x = c. This means the slope of the tangent line at x = c is positive.
Examine each graph to see where the slope of the tangent is zero at x = a and x = b, and positive at x = c. This will help us match the table to the correct graph.
In graph (d), the function has horizontal tangents (f'(x) = 0) at x = a and x = b, and the slope is positive at x = c, matching the conditions given in the table.

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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 is zero or undefined. These points are essential for finding local extrema, as they indicate where the function's slope changes, potentially leading to local maxima or minima. In the given table, points 'a' and 'b' are critical points since their derivatives are zero.
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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 the nature of the extremum. For instance, if the derivative changes from positive to negative at a critical point, it indicates a local maximum.
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The First Derivative Test: Finding Local Extrema

Extrema

Extrema refer to the maximum and minimum values of a function within a given interval. Local extrema are found at critical points, while absolute extrema are the highest or lowest values over the entire domain. In the context of the provided graphs and table, identifying the extrema involves analyzing the critical points and their corresponding function values.
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Finding Extrema Graphically
Related Practice
Textbook Question

Sketch the graphs of the rational functions in Exercises 53–60.


𝓍²

y = ------------------

𝓍² ― 4

Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(x + 1) dx

Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫7sin(θ/3) dθ

Textbook Question

Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = {x² − x, −2 ≤ x ≤−1

2x² − 3x − 3, −1 < x ≤ 0

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Textbook Question

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)

Textbook Question

Finding Functions from Derivatives


Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.