Locate and classify the local extremum of the function f (x, y) x² + y? – 8x – 2y + 6. | -

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
Question
1. Saddle-point at (4, 4)
2. Saddle-point at (4, 1)
3. Local maximum at (1, 1)
4. Local maximum at (4, 1)
5. Local minimum at (4, 1)
Transcribed Image Text:1. Saddle-point at (4, 4) 2. Saddle-point at (4, 1) 3. Local maximum at (1, 1) 4. Local maximum at (4, 1) 5. Local minimum at (4, 1)
**Problem Statement:**

Locate and classify the local extremum of the function 

\[ f(x, y) = x^2 + y^2 - 8x - 2y + 6. \]

**Analysis:**

To find the local extrema of the given function, we need to follow these steps:

1. **Find the Critical Points:**
   - Compute the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \).
   - Set these partial derivatives to zero to find the critical points.

2. **Classify the Critical Points:**
   - Use the second derivative test for functions of two variables to classify each critical point.
   - Compute the second partial derivatives.
   - Calculate the determinant of the Hessian matrix.
   - Use the determinant to determine whether each critical point corresponds to a local minimum, local maximum, or saddle point.  
  
These are the steps involved in locating and classifying the local extrema of the given function of two variables.
Transcribed Image Text:**Problem Statement:** Locate and classify the local extremum of the function \[ f(x, y) = x^2 + y^2 - 8x - 2y + 6. \] **Analysis:** To find the local extrema of the given function, we need to follow these steps: 1. **Find the Critical Points:** - Compute the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \). - Set these partial derivatives to zero to find the critical points. 2. **Classify the Critical Points:** - Use the second derivative test for functions of two variables to classify each critical point. - Compute the second partial derivatives. - Calculate the determinant of the Hessian matrix. - Use the determinant to determine whether each critical point corresponds to a local minimum, local maximum, or saddle point. These are the steps involved in locating and classifying the local extrema of the given function of two variables.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 3 images

Blurred answer
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