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 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

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

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.

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.

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.

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.

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)

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

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)

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap

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.