Using the graph of g 'shown below and the fact that g(0) = 150, find the coordinates of all the critical points and inflection points of g. (20, 10) -10 40 /15 (20 (10,-20) (a) The critical points of g occur at what points? (15 0 g'(x) X) (smaller x value) X) (larger x value) (b) The inflection points of g occur at what points? (10 -20 40 10 X) (smaller x value) X) (larger x value)

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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**Understanding Critical and Inflection Points Using \( g'(x) \)**

This exercise involves analyzing the graph of the derivative \( g'(x) \) to find the critical and inflection points of the function \( g \), given that \( g(0) = 150 \).

### Graph Description:

The graph depicted is of the derivative \( g'(x) \). It shows linear segments connecting the following points:
- Starts at an unspecified point, moves through \( (10, -20) \).
- From \( (10, -20) \), it ascends to \( (15, 0) \).
- Then rises further from \( (15, 0) \) to \( (20, 10) \).
- Finally, it continues to \( (40, 0) \).

### Tasks and Solutions:

#### (a) Critical Points:
The critical points of \( g \) occur where \( g'(x) = 0 \).

- **Answer:** 
  - \( (15, 0) \) and \( (40, 0) \).

#### (b) Inflection Points:
The inflection points of \( g \) occur at locations where the slope of \( g'(x) \) changes, indicating a change in concavity of \( g(x) \).

- **Answer:**
  - \( (10, -20) \) and \( (20, 10) \).

These coordinates provide a guide for determining where the function \( g \) has changes in its slope (critical points) and changes in concavity (inflection points), based on the behavior of its derivative \( g'(x) \).
Transcribed Image Text:**Understanding Critical and Inflection Points Using \( g'(x) \)** This exercise involves analyzing the graph of the derivative \( g'(x) \) to find the critical and inflection points of the function \( g \), given that \( g(0) = 150 \). ### Graph Description: The graph depicted is of the derivative \( g'(x) \). It shows linear segments connecting the following points: - Starts at an unspecified point, moves through \( (10, -20) \). - From \( (10, -20) \), it ascends to \( (15, 0) \). - Then rises further from \( (15, 0) \) to \( (20, 10) \). - Finally, it continues to \( (40, 0) \). ### Tasks and Solutions: #### (a) Critical Points: The critical points of \( g \) occur where \( g'(x) = 0 \). - **Answer:** - \( (15, 0) \) and \( (40, 0) \). #### (b) Inflection Points: The inflection points of \( g \) occur at locations where the slope of \( g'(x) \) changes, indicating a change in concavity of \( g(x) \). - **Answer:** - \( (10, -20) \) and \( (20, 10) \). These coordinates provide a guide for determining where the function \( g \) has changes in its slope (critical points) and changes in concavity (inflection points), based on the behavior of its derivative \( g'(x) \).
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