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On which of the following intervals is f(x)=x4-100x²+x-10 concave up? Selected Answers: A. (-∞,-5) D. (5,00) A. (-∞, -5) B. (-∞, -4) C. (-4,4) D. (5,00) E. (4, ∞ ) Answers:

On which of the following intervals is f(x)=x4-100x²+x-10 concave up? Selected Answers: A. (-∞,-5) D. (5,00) A. (-∞, -5) B. (-∞, -4) C. (-4,4) D. (5,00) E. (4, ∞ ) Answers:

Calculus: Early Transcendentals
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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### Topic: Analyzing Concavity of a Polynomial Function #### Question: On which of the following intervals is \( f(x) = x^4 - 100x^2 + x - 10 \) concave up? #### Provided Options: - A. \((-\infty, -5)\) - B. \((-\infty, -4)\) - C. \((-4, 4)\) - D. \((5, \infty)\) - E. \((4, \infty)\) #### Selected Answers: - A. \((-\infty, -5)\) - D. \((5, \infty)\) #### Correct Answers: - A. \((-\infty, -5)\) - D. \((5, \infty)\) ### Explanation The question involves determining the intervals on which the given polynomial function \( f(x) = x^4 - 100x^2 + x - 10 \) is concave up. **Steps to Analyze Concavity:** 1. **Compute the Second Derivative:** To determine concavity, we need to find the second derivative \( f''(x) \). 2. **Determine Critical Points:** Solve \( f''(x) = 0 \) to find critical points which help identify potential changes in concavity. 3. **Test Intervals:** Use the critical points to test the concavity in each interval. 4. **Identify Concave Up Intervals:** Function \( f(x) \) is concave up where \( f''(x) > 0 \). ### Usage on Educational Platforms: This transcription provides a clear framework for students to understand how to analyze and determine the concavity of polynomial functions using second derivatives. It also includes steps and rationale behind choosing intervals, ensuring that learners can replicate the process independently.
Transcribed Image Text:### Topic: Analyzing Concavity of a Polynomial Function #### Question: On which of the following intervals is \( f(x) = x^4 - 100x^2 + x - 10 \) concave up? #### Provided Options: - A. \((-\infty, -5)\) - B. \((-\infty, -4)\) - C. \((-4, 4)\) - D. \((5, \infty)\) - E. \((4, \infty)\) #### Selected Answers: - A. \((-\infty, -5)\) - D. \((5, \infty)\) #### Correct Answers: - A. \((-\infty, -5)\) - D. \((5, \infty)\) ### Explanation The question involves determining the intervals on which the given polynomial function \( f(x) = x^4 - 100x^2 + x - 10 \) is concave up. **Steps to Analyze Concavity:** 1. **Compute the Second Derivative:** To determine concavity, we need to find the second derivative \( f''(x) \). 2. **Determine Critical Points:** Solve \( f''(x) = 0 \) to find critical points which help identify potential changes in concavity. 3. **Test Intervals:** Use the critical points to test the concavity in each interval. 4. **Identify Concave Up Intervals:** Function \( f(x) \) is concave up where \( f''(x) > 0 \). ### Usage on Educational Platforms: This transcription provides a clear framework for students to understand how to analyze and determine the concavity of polynomial functions using second derivatives. It also includes steps and rationale behind choosing intervals, ensuring that learners can replicate the process independently.
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