Use the intermediate value theorem to determine whether the function f(x) = x^3 + 2x - 4 has a root or not between x =1 and x = 2. If yes, then find the root to five decimal places.
Use the intermediate value theorem to determine whether the function f(x) = x^3 + 2x - 4 has a root or not between x =1 and x = 2. If yes, then find the root to five decimal places.
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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Use the intermediate value theorem to determine whether the function f(x) = x^3 + 2x - 4 has a root or not between x =1 and x = 2. If yes, then find the root to five decimal places.
![### Use of the Intermediate Value Theorem
#### Problem Statement
Use the intermediate value theorem to determine whether the function \( f(x) = x^3 + 2x - 4 \) has a root or not between \( x = 1 \) and \( x = 2 \). If yes, then find the root to five decimal places. Choose the correct answer below:
#### Options
- **Option A**: Yes, the given function has a root between \([1, 2]\) and the root is \(\_\_\_\_\_\) (Type an integer or decimal rounded to five decimal places as needed)
- **Option B**: No, the given function has no root between \([1, 2]\)
---
The problem requires applying the Intermediate Value Theorem, which states that if a function \( f \) is continuous on a closed interval \([a, b]\) and \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F862d597a-ccd1-4202-ba0c-cb7527992470%2F93332037-ef14-419e-90b5-c5c82114db8e%2Fjweatp.jpeg&w=3840&q=75)
Transcribed Image Text:### Use of the Intermediate Value Theorem
#### Problem Statement
Use the intermediate value theorem to determine whether the function \( f(x) = x^3 + 2x - 4 \) has a root or not between \( x = 1 \) and \( x = 2 \). If yes, then find the root to five decimal places. Choose the correct answer below:
#### Options
- **Option A**: Yes, the given function has a root between \([1, 2]\) and the root is \(\_\_\_\_\_\) (Type an integer or decimal rounded to five decimal places as needed)
- **Option B**: No, the given function has no root between \([1, 2]\)
---
The problem requires applying the Intermediate Value Theorem, which states that if a function \( f \) is continuous on a closed interval \([a, b]\) and \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \).
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