5) Use the Intermediate Theorem and Roll's Theorem show that there is exactly one root of the given equation: 3a +x-1=0 between (-1,1)
5) Use the Intermediate Theorem and Roll's Theorem show that there is exactly one root of the given equation: 3a +x-1=0 between (-1,1)
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
![### Exercise: Application of the Intermediate Value Theorem and Rolle's Theorem
**Problem Statement:**
Use the Intermediate Value Theorem and Rolle's Theorem to show that there is exactly one root of the given equation between \((-1,1)\):
\[3x^3 + x - 1 = 0\]
**Explanation:**
- **Intermediate Value Theorem:** This theorem states that if \( f \) is a continuous function on the interval \([a,b]\) and \( N \) is any number between \( f(a) \) and \( f(b) \), then there exists a \( c \) in the interval \((a,b)\) such that \( f(c) = N \).
- **Rolle’s Theorem:** This theorem guarantees that if \( f \) is a continuous function on the closed interval \([a,b]\), differentiable on the open interval \((a,b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in the interval \((a,b)\) such that \( f'(c) = 0 \).
---
**Solution Outline:**
1. **Verify Continuity and Evaluations:**
- The given function \( f(x) = 3x^3 + x - 1 \) is continuous and differentiable everywhere because it is a polynomial.
- Evaluate \( f \) at \( x = -1 \) and \( x = 1 \).
\[
f(-1) = 3(-1)^3 + (-1) - 1 = -3 - 1 - 1 = -5
\]
\[
f(1) = 3(1)^3 + 1 - 1 = 3 + 1 - 1 = 3
\]
2. **Apply the Intermediate Value Theorem:**
- Since \( f(-1) = -5 \) and \( f(1) = 3 \), and zero (0) lies between -5 and 3, by the Intermediate Value Theorem, there exists at least one \( c \) in \((-1,1)\) such that \( f(c) = 0 \).
3. **Apply Rolle’s Theorem:**
- To ensure uniqueness, we check the derivative \( f'(x) \):
\[
f](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb02fb262-35a8-4c2d-87d7-48cdf0e8c8b9%2F9a288867-e697-4043-9aff-3369b44499bd%2Ft3lhace_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exercise: Application of the Intermediate Value Theorem and Rolle's Theorem
**Problem Statement:**
Use the Intermediate Value Theorem and Rolle's Theorem to show that there is exactly one root of the given equation between \((-1,1)\):
\[3x^3 + x - 1 = 0\]
**Explanation:**
- **Intermediate Value Theorem:** This theorem states that if \( f \) is a continuous function on the interval \([a,b]\) and \( N \) is any number between \( f(a) \) and \( f(b) \), then there exists a \( c \) in the interval \((a,b)\) such that \( f(c) = N \).
- **Rolle’s Theorem:** This theorem guarantees that if \( f \) is a continuous function on the closed interval \([a,b]\), differentiable on the open interval \((a,b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in the interval \((a,b)\) such that \( f'(c) = 0 \).
---
**Solution Outline:**
1. **Verify Continuity and Evaluations:**
- The given function \( f(x) = 3x^3 + x - 1 \) is continuous and differentiable everywhere because it is a polynomial.
- Evaluate \( f \) at \( x = -1 \) and \( x = 1 \).
\[
f(-1) = 3(-1)^3 + (-1) - 1 = -3 - 1 - 1 = -5
\]
\[
f(1) = 3(1)^3 + 1 - 1 = 3 + 1 - 1 = 3
\]
2. **Apply the Intermediate Value Theorem:**
- Since \( f(-1) = -5 \) and \( f(1) = 3 \), and zero (0) lies between -5 and 3, by the Intermediate Value Theorem, there exists at least one \( c \) in \((-1,1)\) such that \( f(c) = 0 \).
3. **Apply Rolle’s Theorem:**
- To ensure uniqueness, we check the derivative \( f'(x) \):
\[
f
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, trigonometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning