Identifying Extrema In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2). " style="max-width: 100%; white-space-collapse: preserve;" width="190"> b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.

Welcome back everyone. Suppose the domain of FFX is from -1 up to 1 inclusive. Determine the intervals where F of X is positive and negative, using the graph of its derivative F of X. So let's understand for this problem that the derivative F of X can give us multiple points of information. For example, it can tell us where the function is increasing or decreasing. It can tell us about the concavity. Of the function F of X, it can also give us extreme values, right? Maximum and minimum values. Does the first derivative tell us anything about Function values being positive or negative well. F of X. Does not Tell us About F of X Valleys So just looking at the given graph, we cannot really conclude where F of X is positive or negative. We can sell where it is increasing or decreasing. We can also estimate concavity or possible extreme values, but we cannot really sell where the function itself is positive or negative. So we can conclude for this problem that it cannot be. Determined That will be our final answer to this problem. And thank you for watching.

Table of contents

Skip topic navigation

0. Functions7h 54m

Introduction to Functions
16m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
36m

Exponential & Logarithmic Equations
35m

Introduction to Trigonometric Functions
38m

Graphs of Trigonometric Functions
44m

Trigonometric Identities
47m

Inverse Trigonometric Functions
48m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation3h 18m

Basic Rules of Differentiation
39m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

Derivatives of Trig Functions
40m

The Chain Rule
49m

4. Applications of Derivatives2h 38m

Motion Analysis
36m

Implicit Differentiation
27m

Related Rates
59m

Linearization
13m

Differentials
21m

5. Graphical Applications of Derivatives6h 2m

Intro to Extrema
23m

Finding Global Extrema
50m

The First Derivative Test
1h 6m

Concavity
1h 1m

The Second Derivative Test
31m

Curve Sketching
44m

Applied Optimization
1h 25m

6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m

Derivatives of Exponential & Logarithmic Functions
1h 52m

Logarithmic Differentiation
20m

Derivatives of Inverse Trigonometric Functions
24m

7. Antiderivatives & Indefinite Integrals1h 26m

Antiderivatives
17m

Indefinite Integrals
25m

Integrals of Trig Functions
15m

Initial Value Problems
27m

8. Definite Integrals4h 44m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 21m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
37m

Average Value of a Function
21m

Substitution
1h 19m

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

Integrals of Exponential Functions
40m

Integrals Involving Logarithmic Functions
58m

Integrals Involving Inverse Trigonometric Functions
55m

12. Techniques of Integration7h 41m

Integration by Parts
2h 32m

Partial Fractions
4h 29m

Improper Integrals
39m

13. Intro to Differential Equations2h 55m

Basics of Differential Equations
48m

Slope Fields
19m

Euler's Method
22m

Separable Differential Equations
1h 25m

14. Sequences & Series5h 36m

Sequences
1h 22m

Review of Factorials
11m

Series
44m

Convergence Tests
3h 17m

15. Power Series2h 19m

Introduction to Power Series
1h 9m

Taylor Series & Taylor Polynomials
1h 10m

16. Parametric Equations & Polar Coordinates7h 58m

Parametric Equations
1h 6m

Calculus with Parametric Curves
1h 13m

Polar Coordinates
2h 5m

Calculus in Polar Coordinates
55m

Conic Sections
2h 36m

5. Graphical Applications of Derivatives

Intro to Extrema

Calculus

5. Graphical Applications of Derivatives

Intro to Extrema

Previous problemNext problem

2:06 minutes

Problem 4.3.63b

Textbook Question

Identifying Extrema

In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).

b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.

Verified step by step guidance

Step 1: Observe the graph of f'(x), which represents the derivative of f(x). The derivative indicates the slope of the function f(x). When f'(x) > 0, f(x) is increasing, and when f'(x) < 0, f(x) is decreasing.

Step 2: Identify the intervals where f'(x) is positive (above the x-axis). From the graph, f'(x) is positive on the intervals (-2, -1) and (0, 1). This means f(x) is increasing on these intervals.

Step 3: Identify the intervals where f'(x) is negative (below the x-axis). From the graph, f'(x) is negative on the intervals (-1, 0) and (1, 2). This means f(x) is decreasing on these intervals.

Step 4: Note the points where f'(x) crosses the x-axis. These are the critical points where f'(x) = 0, and f(x) may have local extrema. From the graph, these points occur at x = -1, x = 0, and x = 1.

Step 5: Summarize the behavior of f(x) based on the intervals of positivity and negativity of f'(x). f(x) increases on (-2, -1) and (0, 1), decreases on (-1, 0) and (1, 2), and has potential extrema at x = -1, x = 0, and x = 1.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First Derivative Test

The First Derivative Test is used to determine where a function is increasing or decreasing. If f'(x) > 0, the function f is increasing, and if f'(x) < 0, f is decreasing. By analyzing the sign changes of f'(x), we can identify intervals where f is positive or negative, which helps in finding local extrema.

Critical Points

Critical points occur where the derivative f'(x) is zero or undefined. These points are potential locations for local maxima or minima. In the graph, critical points are where the curve crosses the x-axis, indicating a change in the direction of f(x), which is crucial for identifying extrema.

Domain of the Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this problem, the domain is (-2, 2), meaning we only consider the behavior of f and f' within this interval. Understanding the domain helps focus the analysis on relevant sections of the graph.

Watch next

Master Finding Extrema Graphically with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = sec²x − 2tan x, −π/2 < x < π/2

130

views

Textbook Question

Identifying Extrema

In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.

112

views

Textbook Question

Identifying Extrema

In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.

112

views

Textbook Question

Identifying Extrema

In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).

" style="" width="190">

a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.

114

views

Textbook Question

Identifying Extrema

In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).

" style="" width="220">

a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.

views

Textbook Question

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)

105

views

Textbook Question

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) = (x² − 3) / (x − 2), x ≠ 2

133

views

Textbook Question

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) = x¹ᐟ³(x + 8)

115

views

Show more practices

Download the Mobile app

Accessibility

Privacy

Do not sell my personal information

Identifying ExtremaIn Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.

Key Concepts

First Derivative Test

Critical Points

Domain of the Function

Watch next

Identifying Extrema

In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).

b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.