Locate and classify all the critical points of the function. (If an answer does not exist, enter DNE.) f(x, y) = x² + 2xy + 2y² – 8x + 8y (x, y) = ---Select--- -Select--- relative minimum Submit Answer
Locate and classify all the critical points of the function. (If an answer does not exist, enter DNE.) f(x, y) = x² + 2xy + 2y² – 8x + 8y (x, y) = ---Select--- -Select--- relative minimum Submit Answer
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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![### Critical Points of Functions
In calculus, a critical point of a function \( f(x,y) \) is defined as a point \((x, y)\) where the gradient (first derivatives) \( \nabla f(x, y) = 0 \). Critical points are important because they can help in locating local minima, maxima, or saddle points of the function.
#### Example Problem:
**Locate and classify all the critical points of the function. (If an answer does not exist, enter DNE.)**
\[ f(x, y) = x^2 + 2xy + 2y^2 - 8x + 8y \]
To classify each critical point, you will:
1. **Step 1**: Find the partial derivatives of \( f(x, y) \).
2. **Step 2**: Set the partial derivatives to zero to solve for \( x \) and \( y \).
3. **Step 3**: Identify each critical point and determine its nature (relative minimum, relative maximum, saddle point, or no extremum).
#### Input Format Instructions:
1. **Enter Critical Points**:
- In the given problem, there is an input field to enter the critical points in the format \((x, y) = (\text{ } )\). Ensure that you solve the equations correctly and input all critical points.
2. **Select Classification**:
- There is a drop-down menu to classify the critical point. The options include:
- Relative Minimum
- Relative Maximum
- Saddle Point
- No Extremum
After entering the critical points and selecting the appropriate classification from the drop-down menu, click the "Submit Answer" button to validate your answer.
#### Example Submission:
For a critical point \( (2, -1) \):
- **Point**: \( (2, -1) \)
- **Classification**: Saddle point
#### Additional Tools and Resources:
- **Details Section**: Click the "Details" section to find additional explanations or examples to guide you through similar problems.
- **Error Checking**: Verify your calculations using Lagrange multipliers to ensure there are no computational errors.
Remember that understanding the nature of critical points is fundamental in optimization problems in various fields such as economics, engineering, and physics. Use this template to practice and refine your ability to classify critical points accurately](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e4adde6-a9e7-418d-be79-837ec20737cb%2Fee3c7c75-f9ff-4ea4-8865-a4545c5bbde8%2Fgklcelo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Critical Points of Functions
In calculus, a critical point of a function \( f(x,y) \) is defined as a point \((x, y)\) where the gradient (first derivatives) \( \nabla f(x, y) = 0 \). Critical points are important because they can help in locating local minima, maxima, or saddle points of the function.
#### Example Problem:
**Locate and classify all the critical points of the function. (If an answer does not exist, enter DNE.)**
\[ f(x, y) = x^2 + 2xy + 2y^2 - 8x + 8y \]
To classify each critical point, you will:
1. **Step 1**: Find the partial derivatives of \( f(x, y) \).
2. **Step 2**: Set the partial derivatives to zero to solve for \( x \) and \( y \).
3. **Step 3**: Identify each critical point and determine its nature (relative minimum, relative maximum, saddle point, or no extremum).
#### Input Format Instructions:
1. **Enter Critical Points**:
- In the given problem, there is an input field to enter the critical points in the format \((x, y) = (\text{ } )\). Ensure that you solve the equations correctly and input all critical points.
2. **Select Classification**:
- There is a drop-down menu to classify the critical point. The options include:
- Relative Minimum
- Relative Maximum
- Saddle Point
- No Extremum
After entering the critical points and selecting the appropriate classification from the drop-down menu, click the "Submit Answer" button to validate your answer.
#### Example Submission:
For a critical point \( (2, -1) \):
- **Point**: \( (2, -1) \)
- **Classification**: Saddle point
#### Additional Tools and Resources:
- **Details Section**: Click the "Details" section to find additional explanations or examples to guide you through similar problems.
- **Error Checking**: Verify your calculations using Lagrange multipliers to ensure there are no computational errors.
Remember that understanding the nature of critical points is fundamental in optimization problems in various fields such as economics, engineering, and physics. Use this template to practice and refine your ability to classify critical points accurately
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