Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Finding Global Extrema

Calculus

5. Graphical Applications of Derivatives

Finding Global Extrema

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3:03 minutes

Problem 4.1.7

Textbook Question

Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.

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First, identify the endpoints of the graph. The graph has endpoints at (-1, 1) and (1, -1). These points are important because absolute extrema can occur at endpoints.

Next, observe the graph to identify any local extrema. Local extrema are points where the graph changes direction. In this graph, there is a local maximum at (0, 0) because the graph changes from increasing to decreasing.

Determine the y-values at the endpoints and the local extrema. At (-1, 1), the y-value is 1; at (0, 0), the y-value is 0; and at (1, -1), the y-value is -1.

Compare the y-values to find the absolute maximum and minimum. The absolute maximum is the highest y-value, which is 1 at (-1, 1). The absolute minimum is the lowest y-value, which is -1 at (1, -1).

Conclude that the absolute maximum value is 1 occurring at x = -1, and the absolute minimum value is -1 occurring at x = 1.

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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 graph over a given interval. The absolute maximum is the highest point, while the absolute minimum is the lowest. These points can occur at critical points or endpoints of the interval. Identifying these points involves evaluating the function at critical points and endpoints to determine the largest and smallest values.

Critical Points

Critical points are where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. To find these points, take the derivative of the function and solve for where it equals zero or is undefined. These points are essential in determining where the function's slope changes, which helps in identifying extrema.

Graph Analysis

Graph analysis involves examining a graph to identify key features such as intercepts, slopes, and extrema. By analyzing the graph, one can visually determine where the function reaches its highest or lowest values. This process includes observing the behavior of the graph at endpoints and critical points, which is crucial for finding absolute extrema.

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Related Practice

Textbook Question

Absolute Extrema on Finite Closed Intervals

In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

h(x) = ³√x, −1 ≤ x ≤ 8

Textbook Question

Absolute Extrema on Finite Closed Intervals

g(x) = √(4 − x²), −2 ≤ x ≤ 1

Textbook Question

Absolute Extrema on Finite Closed Intervals

f(t) = 2 − |t|, −1 ≤ t ≤ 3

Textbook Question

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.

Textbook Question

Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.

Textbook Question

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

f(x) = |x|, −1 < x < 2

Textbook Question

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

g(x) = {−x, 0 ≤ x < 1

x − 1, 1 ≤ x ≤ 2

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

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Finding Extrema from GraphsIn Exercises 7–10, find the absolute extreme values and where they occur.

Key Concepts

Absolute Extrema

Critical Points

Graph Analysis

Watch next

Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.