Textbook Question
Theory and Examples
[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.
f(x) = |x − 2| + |x + 3|, −5 ≤ x ≤ 5
212
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.1.5
Finding Extrema from Graphs
In Exercises 1–6, determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with Theorem 1.
Verified step by step guidance1
Examine the graph of the function y = g(x) over the interval [a, b]. Identify any points where the function reaches a maximum or minimum value within this interval.
Notice that the function is continuous on the interval [a, b] except at point c, where there is a discontinuity. The function is not defined at c, as indicated by the open circle.
Identify the endpoints of the interval, a and b, and evaluate the function at these points. These values are potential candidates for absolute extrema.
Look for any local extrema within the interval [a, b] by observing the behavior of the graph. A local maximum or minimum could occur at a point where the graph changes direction.
Apply Theorem 1, which states that if a function is continuous on a closed interval [a, b], it must have an absolute maximum and minimum on that interval. Since the function is not continuous at c, consider only the endpoints and any local extrema that are defined within the interval.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Extrema
Absolute extrema refer to the highest or lowest points on a function within a given interval. An absolute maximum is the highest point, while an absolute minimum is the lowest. These points are crucial in understanding the overall behavior of a function on a specified interval, such as [a, b].
Recommended video:
05:58
Finding Extrema Graphically
Closed Interval
A closed interval [a, b] includes all the points between a and b, as well as the endpoints themselves. This is important when determining extrema because the function must be evaluated at the endpoints to ensure all possible extreme values are considered, especially if the function is continuous.
Recommended video:
02:59Finding Area Between Curves that Cross on the Interval
Theorem 1 (Extreme Value Theorem)
The Extreme Value Theorem states that if a function is continuous on a closed interval [a, b], it must have both a maximum and minimum value on that interval. This theorem is essential for identifying extrema, as it guarantees their existence under the right conditions, guiding the analysis of the graph.
Recommended video:
06:11Fundamental Theorem of Calculus Part 1
Related Practice
Textbook Question
Root Finding
5. Use Newton's method to find the positive fourth root of 2 by solving the equation x^4 -2 = 0. Start with x_0 = 1 and find x_2.
214
views
Textbook Question
Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
71. y' = x(x² - 12)
154
views
Textbook Question
Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.
74. y' = (x² - 2x)(x - 5)²
88
views
Textbook Question
Finding Extreme Values
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = 𝓍³ + 𝓍² ― 8𝓍 + 5
224
views
Textbook Question
119. Find the values of constants a, b, and c such that the graph of y = ax^3 + bx^2 + cx has a
local maximum at x = 3, local minimum at x =- 1, and inflection point at (1, 11).
226
views