Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
Bartleby Related Questions Icon

Related questions

Q: Find the derivative of f(x) = x²(x² + V¤)
A:
Q: Use the graph of the function f(x) to sketch the graph of the derivative f’(x)
A: We have to graph the function f(x) and its derivative.
Q: Dein 96 (2) Calculate the derivative f'(x) of the following function a) f(x) & S 1-x (6) f(x) =…
A:
Q: Is the mathematical statement correct. If f(x)=f(-x), then f’(x)=f’(-x). Explain why
A:
Q: Differentiate the function.
A:
Q: If f(x)=x2 -5x+7, calculate the derivative of f(x). Give it the name f'(x). Compare the domains of…
A: First we have to calculate the derivative of f(x) and then domains
Q: Let f(x) and its linear derivative f'(x) be defined for all real numbers. Use the table of selected…
A: Step 1:Step 2:
Q: f(x) = x(=") – (x*)*, x > 0. – (2*)*, x > 0. %3D
A: We have to find the derivative of  f(x)=x(xx)-(xx)x ,  x>0
Topic Video
Question

let f(x) be a function such that its derivative is given by: f'(x)=5(x-1)^2(x-2)^3(x-3)^4. which statement is true about the function f(x)?

### Analyzing the Derivative of a Function #### Given Problem Let \( f(x) \) be a function such that its derivative is given by: \[ f'(x) = 5(x-1)^2 (x-2)^3 (x-3)^4. \] Which statement is true about the function \( f(x) \)? - \(\bigcirc \) It has three local max or local min at \( x = 1, x = 2, \) and \( x = 3 \) - \(\bigcirc \) It has a local min at \( x = 2 \) - \(\bigcirc \) It has a local min at \( x = 3 \) - \(\bigcirc \) It has no local max nor local min - \(\bigcirc \) None of the above #### Detailed Analysis To determine the local extrema of \( f(x) \), we need to analyze its derivative \( f'(x) \). 1. **Critical Points:** - The derivative \( f'(x) = 0 \) at \( x = 1, 2, 3 \). 2. **Behavior of \( f'(x) \) Based on its Factors:** - **At \( x = 1 \):** The factor \((x-1)^2\) makes \( f'(x) \) equal zero. Since the power is even (2), the sign of \( f'(x) \) does not change around \( x = 1 \). This suggests that \( x = 1 \) is more likely a point of inflection rather than a local maximum or minimum. - **At \( x = 2 \):** The factor \((x-2)^3\) makes \( f'(x) \) equal zero. Since the power is odd (3), the sign of \( f'(x) \) changes around \( x = 2 \). This indicates that \( x = 2 \) is a local extremum. Given that the factor is raised to an odd power greater than 1, which affects the steepness at that point, it suggests a local minimum. - **At \( x = 3 \):** The factor \((x-3)^4\) makes \( f'(x) \) equal zero. Since the power is even (4),
expand button
Transcribed Image Text:### Analyzing the Derivative of a Function #### Given Problem Let \( f(x) \) be a function such that its derivative is given by: \[ f'(x) = 5(x-1)^2 (x-2)^3 (x-3)^4. \] Which statement is true about the function \( f(x) \)? - \(\bigcirc \) It has three local max or local min at \( x = 1, x = 2, \) and \( x = 3 \) - \(\bigcirc \) It has a local min at \( x = 2 \) - \(\bigcirc \) It has a local min at \( x = 3 \) - \(\bigcirc \) It has no local max nor local min - \(\bigcirc \) None of the above #### Detailed Analysis To determine the local extrema of \( f(x) \), we need to analyze its derivative \( f'(x) \). 1. **Critical Points:** - The derivative \( f'(x) = 0 \) at \( x = 1, 2, 3 \). 2. **Behavior of \( f'(x) \) Based on its Factors:** - **At \( x = 1 \):** The factor \((x-1)^2\) makes \( f'(x) \) equal zero. Since the power is even (2), the sign of \( f'(x) \) does not change around \( x = 1 \). This suggests that \( x = 1 \) is more likely a point of inflection rather than a local maximum or minimum. - **At \( x = 2 \):** The factor \((x-2)^3\) makes \( f'(x) \) equal zero. Since the power is odd (3), the sign of \( f'(x) \) changes around \( x = 2 \). This indicates that \( x = 2 \) is a local extremum. Given that the factor is raised to an odd power greater than 1, which affects the steepness at that point, it suggests a local minimum. - **At \( x = 3 \):** The factor \((x-3)^4\) makes \( f'(x) \) equal zero. Since the power is even (4),
Expert Solution
Check Mark
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
bartleby
This is a popular solution
See solutionCheck out a sample Q&A here
Step 1
VIEW
Step 2
VIEW
bartleby

Trending nowThis is a popular solution!

bartleby

Step by stepSolved in 2 steps with 3 images

See solution
Check out a sample Q&A here
Blurred answer
Knowledge Booster
Background pattern image
Calculus
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.
Similar questions
Recommended textbooks for you
Text book image
Calculus: Early Transcendentals
Calculus
ISBN:9781285741550
Author:James Stewart
Publisher:Cengage Learning
Text book image
Thomas' Calculus (14th Edition)
Calculus
ISBN:9780134438986
Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:PEARSON
Text book image
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:9780134763644
Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:PEARSON
Text book image
Calculus: Early Transcendentals
Calculus
ISBN:9781319050740
Author:Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:W. H. Freeman
Text book image
Precalculus
Calculus
ISBN:9780135189405
Author:Michael Sullivan
Publisher:PEARSON
Text book image
Calculus: Early Transcendental Functions
Calculus
ISBN:9781337552516
Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning
SEE MORE TEXTBOOKS