Calculus: Early Transcendentals
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Author: James Stewart
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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.](https://content.bartleby.com/qna-images/question/ec7ca5ac-d913-4f83-b702-3cb8fd792592/3435f5b3-4def-45c3-9777-1427225f15c1/dxypei_processed.png)
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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