Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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### Problem Statement 5. Let \( f(x) = 4x - 2x^3 - 1 \). Explain, by using the Intermediate Value Theorem, why this function has (at least) three x-intercepts in the interval \((-3, 4)\). *(Note: The instruction regarding credit underneath is obscured and cannot be transcribed.)* ### Explanation The Intermediate Value Theorem states that for any continuous function \( f \), if \( f(a) \) and \( f(b) \) are of opposite signs (i.e., one is positive, and the other is negative), then there is at least one value \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \). #### Steps for Explanation: 1. **Continuity**: Confirm that the function \( f(x) = 4x - 2x^3 - 1 \) is continuous over the interval \((-3, 4)\). The function is a polynomial, and polynomials are continuous everywhere. 2. **Evaluate Endpoints**: - Calculate \( f(-3) \) and \( f(4) \). - Determine the signs of these values. If they are opposite in sign, then the Intermediate Value Theorem guarantees at least one root in that interval. 3. **Evaluate Intermediate Points**: - You may need to find additional points in the interval to guarantee the existence of at least three x-intercepts. This involves checking values and locating additional sign changes. Through this method, the required explanation for the existence of at least three x-intercepts can be derived.
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Transcribed Image Text:### Problem Statement 5. Let \( f(x) = 4x - 2x^3 - 1 \). Explain, by using the Intermediate Value Theorem, why this function has (at least) three x-intercepts in the interval \((-3, 4)\). *(Note: The instruction regarding credit underneath is obscured and cannot be transcribed.)* ### Explanation The Intermediate Value Theorem states that for any continuous function \( f \), if \( f(a) \) and \( f(b) \) are of opposite signs (i.e., one is positive, and the other is negative), then there is at least one value \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \). #### Steps for Explanation: 1. **Continuity**: Confirm that the function \( f(x) = 4x - 2x^3 - 1 \) is continuous over the interval \((-3, 4)\). The function is a polynomial, and polynomials are continuous everywhere. 2. **Evaluate Endpoints**: - Calculate \( f(-3) \) and \( f(4) \). - Determine the signs of these values. If they are opposite in sign, then the Intermediate Value Theorem guarantees at least one root in that interval. 3. **Evaluate Intermediate Points**: - You may need to find additional points in the interval to guarantee the existence of at least three x-intercepts. This involves checking values and locating additional sign changes. Through this method, the required explanation for the existence of at least three x-intercepts can be derived.
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