t g(x) = 2x² + 6x – 4 in the interval [-1,1]. Use the Intermediate Value Theorem to determine if a ot exists for the function in the given interval. esure to justify your answer.) O Yes, a root exists. O No, a root does not exist. O Not Enough Information
t g(x) = 2x² + 6x – 4 in the interval [-1,1]. Use the Intermediate Value Theorem to determine if a ot exists for the function in the given interval. esure to justify your answer.) O Yes, a root exists. O No, a root does not exist. O Not Enough Information
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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![**Problem Statement:**
Let \( g(x) = 2x^3 + 6x - 4 \) in the interval \([-1, 1]\). Use the Intermediate Value Theorem to determine if a root exists for the function in the given interval.
(Be sure to justify your answer.)
- ⓐ Yes, a root exists.
- ⓑ No, a root does not exist.
- ⓒ Not Enough Information
**Explanation:**
The Intermediate Value Theorem states that if a function \( g \) is continuous on the interval \([a, b]\) and if \( g(a) \) and \( g(b) \) have opposite signs, then there exists at least one \( c \) in the interval \((a, b)\) such that \( g(c) = 0 \).
To determine if a root exists for the function in the interval \([-1, 1]\):
1. Evaluate \( g(-1) \).
2. Evaluate \( g(1) \).
3. Check if \( g(-1) \) and \( g(1) \) have opposite signs. If they do, a root must exist within the interval \([-1, 1]\).
Perform these evaluations and use the results to choose the appropriate answer from the given choices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec7ca5ac-d913-4f83-b702-3cb8fd792592%2F3435f5b3-4def-45c3-9777-1427225f15c1%2Fdxypei_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( g(x) = 2x^3 + 6x - 4 \) in the interval \([-1, 1]\). Use the Intermediate Value Theorem to determine if a root exists for the function in the given interval.
(Be sure to justify your answer.)
- ⓐ Yes, a root exists.
- ⓑ No, a root does not exist.
- ⓒ Not Enough Information
**Explanation:**
The Intermediate Value Theorem states that if a function \( g \) is continuous on the interval \([a, b]\) and if \( g(a) \) and \( g(b) \) have opposite signs, then there exists at least one \( c \) in the interval \((a, b)\) such that \( g(c) = 0 \).
To determine if a root exists for the function in the interval \([-1, 1]\):
1. Evaluate \( g(-1) \).
2. Evaluate \( g(1) \).
3. Check if \( g(-1) \) and \( g(1) \) have opposite signs. If they do, a root must exist within the interval \([-1, 1]\).
Perform these evaluations and use the results to choose the appropriate answer from the given choices.
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