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It is known that for a function f(x), f'(1) = 0 and f" (1) = -0.5. What does the Second Derivative Test say, if anything, about the function at x = 1? O There is a relative max at x = 1. O There is a relative min at x = 1. O There is a point of inflection at x = 1. O The test is inconclusive at x = 1. O There is a singular point at x = 1.

It is known that for a function f(x), f'(1) = 0 and f" (1) = -0.5. What does the Second Derivative Test say, if anything, about the function at x = 1? O There is a relative max at x = 1. O There is a relative min at x = 1. O There is a point of inflection at x = 1. O The test is inconclusive at x = 1. O There is a singular point at x = 1.

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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**Understanding the Second Derivative Test** **Question:** It is known that for a function \( f(x) \), \( f'(1) = 0 \) and \( f''(1) = -0.5 \). What does the Second Derivative Test say, if anything, about the function at \( x = 1 \)? **Options:** 1. **There is a relative max at \( x = 1 \).** 2. **There is a relative min at \( x = 1 \).** 3. **There is a point of inflection at \( x = 1 \).** 4. **The test is inconclusive at \( x = 1 \).** 5. **There is a singular point at \( x = 1 \).** ### Explanation of Each Option: - **Option 1: There is a relative max at \( x = 1 \).** This would be true if \( f'(1) = 0 \) and \( f''(1) < 0 \). Since \( f''(1) = -0.5 \), which is less than zero, this option might be correct. - **Option 2: There is a relative min at \( x = 1 \).** This would be true if \( f'(1) = 0 \) and \( f''(1) > 0 \). However, \( f''(1) = -0.5 \), which is not greater than zero, so this option is incorrect. - **Option 3: There is a point of inflection at \( x = 1 \).** A point of inflection occurs where the second derivative changes sign, not just where \( f'(x) = 0 \). This option is incorrect as it does not describe the scenario provided. - **Option 4: The test is inconclusive at \( x = 1 \).** This would be true if \( f'(1) = 0 \) and \( f''(1) = 0 \). Since \( f''(1) = -0.5 \), the test is not inconclusive, so this option is incorrect. - **Option 5: There is a singular point at \( x = 1 \).** A singular point implies that the function is not differentiable at \( x = 1 \), which contradicts the
Transcribed Image Text:**Understanding the Second Derivative Test** **Question:** It is known that for a function \( f(x) \), \( f'(1) = 0 \) and \( f''(1) = -0.5 \). What does the Second Derivative Test say, if anything, about the function at \( x = 1 \)? **Options:** 1. **There is a relative max at \( x = 1 \).** 2. **There is a relative min at \( x = 1 \).** 3. **There is a point of inflection at \( x = 1 \).** 4. **The test is inconclusive at \( x = 1 \).** 5. **There is a singular point at \( x = 1 \).** ### Explanation of Each Option: - **Option 1: There is a relative max at \( x = 1 \).** This would be true if \( f'(1) = 0 \) and \( f''(1) < 0 \). Since \( f''(1) = -0.5 \), which is less than zero, this option might be correct. - **Option 2: There is a relative min at \( x = 1 \).** This would be true if \( f'(1) = 0 \) and \( f''(1) > 0 \). However, \( f''(1) = -0.5 \), which is not greater than zero, so this option is incorrect. - **Option 3: There is a point of inflection at \( x = 1 \).** A point of inflection occurs where the second derivative changes sign, not just where \( f'(x) = 0 \). This option is incorrect as it does not describe the scenario provided. - **Option 4: The test is inconclusive at \( x = 1 \).** This would be true if \( f'(1) = 0 \) and \( f''(1) = 0 \). Since \( f''(1) = -0.5 \), the test is not inconclusive, so this option is incorrect. - **Option 5: There is a singular point at \( x = 1 \).** A singular point implies that the function is not differentiable at \( x = 1 \), which contradicts the
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