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...
Related questions
Question
How do I get fxx, fyy, and fxy?
![**Review: Find and classify CPs (Critical Points) of \( f(x, y) = e^{x^2 + y^2 - 2x} \)**
1. Compute the partial derivative with respect to \( x \):
\[
f_x = (2x - 2) e^{x^2 + y^2 - 2x}
\]
Setting \( f_x = 0 \), gives \( x = 1 \).
*Note: \( e^{\text{any #}} > 0 \)*
2. Compute the partial derivative with respect to \( y \):
\[
f_y = (2y) e^{x^2 + y^2 - 2x}
\]
Setting \( f_y = 0 \), gives \( y = 0 \).
3. The critical point is \( CP = (1, 0) \).
4. Second partial derivatives:
\[
f_{xx} = 2e^{x^2 + y^2 - 2x} + (2x - 2)^2 e^{x^2 + y^2 - 2x}
\]
\[
f_{yy} = 2e^{x^2 + y^2 - 2x} + 4y^2 e^{x^2 + y^2 - 2x}
\]
\[
f_{xy} = 2y(2x - 2) e^{x^2 + y^2 - 2x}
\]
5. Evaluate the Hessian determinant \( H \) at the critical point \( (1, 0) \):
\[
H(1, 0) = 2e^{-1}(2e^{-1}) - 0^2 > 0
\]
Additionally, check \( f_{xx}(1, 0) > 0 \):
\[
f_{xx}(1, 0) = 2e^{-1} > 0
\]
6. Conclusion: Relative minimum at \( (1, 0) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7cce1808-caf5-457b-bc5a-f7c079ff1a3b%2Fbda997eb-5da0-4da1-808c-89806340c478%2Fbfar7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Review: Find and classify CPs (Critical Points) of \( f(x, y) = e^{x^2 + y^2 - 2x} \)**
1. Compute the partial derivative with respect to \( x \):
\[
f_x = (2x - 2) e^{x^2 + y^2 - 2x}
\]
Setting \( f_x = 0 \), gives \( x = 1 \).
*Note: \( e^{\text{any #}} > 0 \)*
2. Compute the partial derivative with respect to \( y \):
\[
f_y = (2y) e^{x^2 + y^2 - 2x}
\]
Setting \( f_y = 0 \), gives \( y = 0 \).
3. The critical point is \( CP = (1, 0) \).
4. Second partial derivatives:
\[
f_{xx} = 2e^{x^2 + y^2 - 2x} + (2x - 2)^2 e^{x^2 + y^2 - 2x}
\]
\[
f_{yy} = 2e^{x^2 + y^2 - 2x} + 4y^2 e^{x^2 + y^2 - 2x}
\]
\[
f_{xy} = 2y(2x - 2) e^{x^2 + y^2 - 2x}
\]
5. Evaluate the Hessian determinant \( H \) at the critical point \( (1, 0) \):
\[
H(1, 0) = 2e^{-1}(2e^{-1}) - 0^2 > 0
\]
Additionally, check \( f_{xx}(1, 0) > 0 \):
\[
f_{xx}(1, 0) = 2e^{-1} > 0
\]
6. Conclusion: Relative minimum at \( (1, 0) \).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
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
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
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
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning