Describe the graph of f at the given point relative to the existence of a local maximum or minimum. Assume that f(x) is continuous on (- 00, co). (5,f(5)) if f'(5) = 0 and f'"(5) > 0 ..... What is the best description of the graph of f at the point (5,f(5))? O A. Unable to determine from the given information O B. Neither O C. Local maximum O D. Local minimum

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:**

Describe the graph of \( f \) at the given point relative to the existence of a local maximum or minimum. Assume that \( f(x) \) is continuous on \((- \infty, \infty)\).

Given:
- \( f'(5) = 0 \)
- \( f''(5) > 0 \)

**What is the best description of the graph of \( f \) at the point \( (5, f(5)) \)?**

- A. Unable to determine from the given information
- B. Neither
- C. Local maximum
- D. Local minimum

**Explanation:**

The question prompts the reader to analyze the behavior of the function \( f \) at the point \( (5, f(5)) \) based on the first and second derivative tests.

- The first derivative \( f'(5) = 0 \) indicates that there is a critical point at \( x = 5 \) which could be a local maximum, local minimum, or a saddle point.
- The second derivative \( f''(5) > 0 \) suggests that the function is concave up at \( x = 5 \).

Based on these conditions, the correct answer is:

- D. Local minimum

This is because a positive second derivative at a critical point indicates that the function is curving upwards, suggesting a local minimum at that point.
Transcribed Image Text:**Question:** Describe the graph of \( f \) at the given point relative to the existence of a local maximum or minimum. Assume that \( f(x) \) is continuous on \((- \infty, \infty)\). Given: - \( f'(5) = 0 \) - \( f''(5) > 0 \) **What is the best description of the graph of \( f \) at the point \( (5, f(5)) \)?** - A. Unable to determine from the given information - B. Neither - C. Local maximum - D. Local minimum **Explanation:** The question prompts the reader to analyze the behavior of the function \( f \) at the point \( (5, f(5)) \) based on the first and second derivative tests. - The first derivative \( f'(5) = 0 \) indicates that there is a critical point at \( x = 5 \) which could be a local maximum, local minimum, or a saddle point. - The second derivative \( f''(5) > 0 \) suggests that the function is concave up at \( x = 5 \). Based on these conditions, the correct answer is: - D. Local minimum This is because a positive second derivative at a critical point indicates that the function is curving upwards, suggesting a local minimum at that point.
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