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How many inflection points does the function f(x)=x²(x° – 1) d. 6 have? а. 1 b. 4 с. 5 e. none

How many inflection points does the function f(x)=x²(x° – 1) d. 6 have? а. 1 b. 4 с. 5 e. none

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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**Question:** How many inflection points does the function \( f(x) = -x^2(x^6 - 1) \) have? a. 1 b. 4 c. 5 d. 6 e. none **Explanation:** To determine the number of inflection points of the function \( f(x) = -x^2(x^6 - 1) \), one would typically follow these steps: 1. **Find the Second Derivative:** Calculate the first derivative \( f'(x) \) and then compute the second derivative \( f''(x) \). 2. **Set the Second Derivative to Zero:** Solve \( f''(x) = 0 \) to find potential inflection points. 3. **Test Intervals Around Solutions:** Verify changes in concavity by checking the sign of \( f''(x) \) in intervals around the solutions. An inflection point occurs where the concavity changes from concave up to concave down or vice versa. The response should be determined through these steps, ensuring a thorough process of calculation to identify how many points conform to the conditions of being inflection points.
Transcribed Image Text:**Question:** How many inflection points does the function \( f(x) = -x^2(x^6 - 1) \) have? a. 1 b. 4 c. 5 d. 6 e. none **Explanation:** To determine the number of inflection points of the function \( f(x) = -x^2(x^6 - 1) \), one would typically follow these steps: 1. **Find the Second Derivative:** Calculate the first derivative \( f'(x) \) and then compute the second derivative \( f''(x) \). 2. **Set the Second Derivative to Zero:** Solve \( f''(x) = 0 \) to find potential inflection points. 3. **Test Intervals Around Solutions:** Verify changes in concavity by checking the sign of \( f''(x) \) in intervals around the solutions. An inflection point occurs where the concavity changes from concave up to concave down or vice versa. The response should be determined through these steps, ensuring a thorough process of calculation to identify how many points conform to the conditions of being inflection points.
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