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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![### Finding the Absolute Extremum within the Specified Domain
In the given problems, we are asked to determine the absolute extremum (either minimum or maximum) of the provided functions over the specified domain. Here, we have three different functions to analyze:
#### Problem 21:
**Find the Minimum of \( f(x) = (x^2 + 4)^{2/3} \) in the interval \([-2, 2]\)**
#### Problem 22:
**Find the Maximum of \( f(x) = 3x^4 + 16x^3 + 24x^2 + 32 \) in the interval \([-3, 1]\)**
#### Problem 23:
**Find the Minimum of \( f(x) = x^3 - 3x^2 \) in the interval \([-0.5, 4]\)**
### Explanation of Graphs or Diagrams
In the current text and format provided, there are no graphs or diagrams included. However, a typical approach to solving these problems may involve plotting the function or analyzing features of the function, such as critical points, to determine the extremum values within the given domains.
**Problem 21 Analysis:**
To find the minimum within \([-2, 2]\), we examine the function \( f(x) = (x^2 + 4)^{2/3} \). This function derives from a polynomial \( x^2 + 4 \), which is always positive, shifted by an exponent. By computing the derivative and setting it to zero, we can find critical points and then determine the minimum within the specified interval.
**Problem 22 Analysis:**
To ascertain the maximum within \([-3, 1]\), we focus on the function \( f(x) = 3x^4 + 16x^3 + 24x^2 + 32 \). Given its polynomial structure, critical points will occur where the first derivative equals zero. Checking these points along with the boundary values will highlight the maximum value within the interval.
**Problem 23 Analysis:**
For finding the minimum within \([-0.5, 4]\), the function \( f(x) = x^3 - 3x^2 \) needs assessment. By taking the first derivative and identifying critical points, and comparing these with the boundary points, the minimum value in the interval can be determined.
Using graphical tools helps visualize these](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b6dc6c4-2828-4529-a1c7-3495f006d86b%2F8387c0b3-914d-410f-8925-1bb6ebec5f9e%2Fum5hi8m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Absolute Extremum within the Specified Domain
In the given problems, we are asked to determine the absolute extremum (either minimum or maximum) of the provided functions over the specified domain. Here, we have three different functions to analyze:
#### Problem 21:
**Find the Minimum of \( f(x) = (x^2 + 4)^{2/3} \) in the interval \([-2, 2]\)**
#### Problem 22:
**Find the Maximum of \( f(x) = 3x^4 + 16x^3 + 24x^2 + 32 \) in the interval \([-3, 1]\)**
#### Problem 23:
**Find the Minimum of \( f(x) = x^3 - 3x^2 \) in the interval \([-0.5, 4]\)**
### Explanation of Graphs or Diagrams
In the current text and format provided, there are no graphs or diagrams included. However, a typical approach to solving these problems may involve plotting the function or analyzing features of the function, such as critical points, to determine the extremum values within the given domains.
**Problem 21 Analysis:**
To find the minimum within \([-2, 2]\), we examine the function \( f(x) = (x^2 + 4)^{2/3} \). This function derives from a polynomial \( x^2 + 4 \), which is always positive, shifted by an exponent. By computing the derivative and setting it to zero, we can find critical points and then determine the minimum within the specified interval.
**Problem 22 Analysis:**
To ascertain the maximum within \([-3, 1]\), we focus on the function \( f(x) = 3x^4 + 16x^3 + 24x^2 + 32 \). Given its polynomial structure, critical points will occur where the first derivative equals zero. Checking these points along with the boundary values will highlight the maximum value within the interval.
**Problem 23 Analysis:**
For finding the minimum within \([-0.5, 4]\), the function \( f(x) = x^3 - 3x^2 \) needs assessment. By taking the first derivative and identifying critical points, and comparing these with the boundary points, the minimum value in the interval can be determined.
Using graphical tools helps visualize these
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