our n en St Suppose that f(x, y) = x² - xy + y²-5x + 5y with x² + y² ≤ 25. 1. Absolute minimum of f(x, y) is 2. Absolute maximum is Submit Question
our n en St Suppose that f(x, y) = x² - xy + y²-5x + 5y with x² + y² ≤ 25. 1. Absolute minimum of f(x, y) is 2. Absolute maximum is Submit Question
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Calculus Practice Problem
Suppose that \( f(x, y) = x^2 - xy + y^2 - 5x + 5y \) with \( x^2 + y^2 \leq 25 \).
1. Absolute minimum of \( f(x, y) \) is: \_\_\_\_\_\_\_\_\_\_\_\_
2. Absolute maximum of \( f(x, y) \) is: \_\_\_\_\_\_\_\_\_\_\_\_
[**Submit Question**]
### Explanation
This problem involves finding the absolute minimum and maximum values of a multivariable function \( f(x, y) \) within a given domain. The domain is specified by the inequality \( x^2 + y^2 \leq 25 \), which describes a circle centered at the origin with radius 5.
#### Steps to Solve:
1. **Identify the Critical Points:**
- Set the partial derivatives \( f_x \) and \( f_y \) equal to zero and solve for \( x \) and \( y \).
2. **Evaluate the Function at Critical Points:**
- Substitute these points into the function \( f(x, y) \) to find the function values.
3. **Examine the Boundary:**
- Parameterize the boundary \( x^2 + y^2 = 25 \) using \( x = 5 \cos(t) \) and \( y = 5 \sin(t) \). Evaluate \( f(5 \cos(t), 5 \sin(t)) \) for \( t \in [0, 2\pi] \).
4. **Compare Values:**
- Compare the function values obtained from the critical points and the boundary to determine the absolute minimum and maximum values.
Use this information to fill in the blanks for the absolute minimum and maximum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd2bc9f06-7337-4a04-9aea-9ce8610fc600%2F594a6d62-c9c5-4e0c-9550-6a0ee8be1b75%2Fn8m1rimo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculus Practice Problem
Suppose that \( f(x, y) = x^2 - xy + y^2 - 5x + 5y \) with \( x^2 + y^2 \leq 25 \).
1. Absolute minimum of \( f(x, y) \) is: \_\_\_\_\_\_\_\_\_\_\_\_
2. Absolute maximum of \( f(x, y) \) is: \_\_\_\_\_\_\_\_\_\_\_\_
[**Submit Question**]
### Explanation
This problem involves finding the absolute minimum and maximum values of a multivariable function \( f(x, y) \) within a given domain. The domain is specified by the inequality \( x^2 + y^2 \leq 25 \), which describes a circle centered at the origin with radius 5.
#### Steps to Solve:
1. **Identify the Critical Points:**
- Set the partial derivatives \( f_x \) and \( f_y \) equal to zero and solve for \( x \) and \( y \).
2. **Evaluate the Function at Critical Points:**
- Substitute these points into the function \( f(x, y) \) to find the function values.
3. **Examine the Boundary:**
- Parameterize the boundary \( x^2 + y^2 = 25 \) using \( x = 5 \cos(t) \) and \( y = 5 \sin(t) \). Evaluate \( f(5 \cos(t), 5 \sin(t)) \) for \( t \in [0, 2\pi] \).
4. **Compare Values:**
- Compare the function values obtained from the critical points and the boundary to determine the absolute minimum and maximum values.
Use this information to fill in the blanks for the absolute minimum and maximum.
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