**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.

Answered: Image /qna-images/answer/3435f5b3-4def-45c3-9777-1427225f15c1.jpg

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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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

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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.

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