5- 4 -5 -4 -3 -2 I 2 3 -4 -5 At the point shown on the function above, which of the following is true? f' < 0, f'' < 0 f' > 0, f'' > 0 | f' < 0, ƒ'' > 0 Of' > 0, ƒ'' < 0

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...
icon
Related questions
icon
Concept explainers
Question
### Understanding Concavity and Derivative Signs

Below is a graph of a function alongside a multiple-choice question related to calculus concepts, specifically concerning the first and second derivatives at a given point.

#### Graph Explanation:

The graph displays a function \( f(x) \) plotted on a standard Cartesian coordinate system with the x-axis ranging from -5 to 5 and the y-axis ranging from -5 to 5. The function has critical points and inflection points illustrating where the slope of the tangent line (first derivative) and the concavity (second derivative) change.

**Key Points on the Graph:**
- The graph exhibits a peak around \( x = -3 \).
- The function then decreases, passing through the x-axis near \( x = -1 \).
- A local minimum at approximately \( x = 4 \) is denoted by a black dot on the graph.

#### Question:

*At the point shown on the function above, which of the following is true?*

1. \(\ f' < 0, f'' < 0 \)
2. \(\ f' > 0, f'' > 0 \)
3. \(\ f' < 0, f'' > 0 \)
4. \(\ f' > 0, f'' < 0 \)

#### Detailed Analysis:

- **At the marked point (black dot),** the function is at its local minimum around \( x = 4 \). 

  - **First Derivative \((f')**: 
    - The first derivative \( f'(x) \) represents the slope of the tangent line at a given point on the function. Since the point is a local minimum, the slope \( f'(x) \) at this point is \( 0 \).
  
  - **Second Derivative \((f'')**: 
    - The second derivative \( f''(x) \) indicates the concavity of the function. At a local minimum, the concavity of the function is upwards, suggesting that the second derivative is positive \( (f''(x) > 0) \).

Given that the point is at a local minimum, the correct answer regarding the concavity and derivative signs at this point is:

\(\ f' < 0, f'' > 0 \) 

**Note:** Considering the selected choices, none exactly match the derivative \( f' = 0 \) and \( f
Transcribed Image Text:### Understanding Concavity and Derivative Signs Below is a graph of a function alongside a multiple-choice question related to calculus concepts, specifically concerning the first and second derivatives at a given point. #### Graph Explanation: The graph displays a function \( f(x) \) plotted on a standard Cartesian coordinate system with the x-axis ranging from -5 to 5 and the y-axis ranging from -5 to 5. The function has critical points and inflection points illustrating where the slope of the tangent line (first derivative) and the concavity (second derivative) change. **Key Points on the Graph:** - The graph exhibits a peak around \( x = -3 \). - The function then decreases, passing through the x-axis near \( x = -1 \). - A local minimum at approximately \( x = 4 \) is denoted by a black dot on the graph. #### Question: *At the point shown on the function above, which of the following is true?* 1. \(\ f' < 0, f'' < 0 \) 2. \(\ f' > 0, f'' > 0 \) 3. \(\ f' < 0, f'' > 0 \) 4. \(\ f' > 0, f'' < 0 \) #### Detailed Analysis: - **At the marked point (black dot),** the function is at its local minimum around \( x = 4 \). - **First Derivative \((f')**: - The first derivative \( f'(x) \) represents the slope of the tangent line at a given point on the function. Since the point is a local minimum, the slope \( f'(x) \) at this point is \( 0 \). - **Second Derivative \((f'')**: - The second derivative \( f''(x) \) indicates the concavity of the function. At a local minimum, the concavity of the function is upwards, suggesting that the second derivative is positive \( (f''(x) > 0) \). Given that the point is at a local minimum, the correct answer regarding the concavity and derivative signs at this point is: \(\ f' < 0, f'' > 0 \) **Note:** Considering the selected choices, none exactly match the derivative \( f' = 0 \) and \( f
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Knowledge Booster
Fundamentals of Algebraic Equations
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning