Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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2a. 2b.
![**Title: Analyzing Critical Points and Local Minima from a Derivative Graph**
**Content:**
2) Suppose \( g \) is a continuous function on the interval \((-5, 4)\). The graph of \( g' \), the derivative of \( g \), is given in the figure below.
**Graph Analysis:**
The graph displays the derivative \( g' \) of the function \( g \) over the interval \((-5, 4)\). The graph can be described as follows:
- From \( x = -5 \) to \( x = -3 \), the graph rises, crossing the x-axis at \( x = -4 \).
- From \( x = -3 \) to \( x = 1 \), the graph decreases, crossing the x-axis at \( x = -1 \).
- From \( x = 1 \) to \( x = 4 \), the graph increases again, crossing the x-axis at \( x = 3 \).
Noticeable points on the graph include:
- Peaks or valleys where the curve changes direction (possible critical points).
- Intervals where the derivative \( g' \) is positive (function \( g \) increases).
- Intervals where the derivative \( g' \) is negative (function \( g \) decreases).
**Exercises:**
a) **List all the critical points (if any) of \( g \) in the interval \((-5, 4)\). Write "NONE" if appropriate.**
\[
x = \_\_\_\_\_\_\_\_\_\_\_\_
\]
b) **At what point (or points) does the function \( g \) have a local minimum? Write "NONE" if appropriate.**
\[
x = \_\_\_\_\_\_\_\_\_\_\_\_
\]
**Guidance for Answering:**
- **Critical Points:** Critical points occur where the derivative \( g' \) is zero or undefined. Observe where the graph crosses the x-axis.
- **Local Minima:** A local minimum can occur at a critical point where \( g' \) changes from negative to positive. Analyze the intervals of positivity and negativity in the graph of \( g' \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7323f76-5a9c-46b6-9930-86cefe6607d1%2Fe099d5a2-e8a4-44f5-8892-7f9a7229d700%2Fhuot5s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Analyzing Critical Points and Local Minima from a Derivative Graph**
**Content:**
2) Suppose \( g \) is a continuous function on the interval \((-5, 4)\). The graph of \( g' \), the derivative of \( g \), is given in the figure below.
**Graph Analysis:**
The graph displays the derivative \( g' \) of the function \( g \) over the interval \((-5, 4)\). The graph can be described as follows:
- From \( x = -5 \) to \( x = -3 \), the graph rises, crossing the x-axis at \( x = -4 \).
- From \( x = -3 \) to \( x = 1 \), the graph decreases, crossing the x-axis at \( x = -1 \).
- From \( x = 1 \) to \( x = 4 \), the graph increases again, crossing the x-axis at \( x = 3 \).
Noticeable points on the graph include:
- Peaks or valleys where the curve changes direction (possible critical points).
- Intervals where the derivative \( g' \) is positive (function \( g \) increases).
- Intervals where the derivative \( g' \) is negative (function \( g \) decreases).
**Exercises:**
a) **List all the critical points (if any) of \( g \) in the interval \((-5, 4)\). Write "NONE" if appropriate.**
\[
x = \_\_\_\_\_\_\_\_\_\_\_\_
\]
b) **At what point (or points) does the function \( g \) have a local minimum? Write "NONE" if appropriate.**
\[
x = \_\_\_\_\_\_\_\_\_\_\_\_
\]
**Guidance for Answering:**
- **Critical Points:** Critical points occur where the derivative \( g' \) is zero or undefined. Observe where the graph crosses the x-axis.
- **Local Minima:** A local minimum can occur at a critical point where \( g' \) changes from negative to positive. Analyze the intervals of positivity and negativity in the graph of \( g' \).
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