Find all absolute extrema of each function. Enter each point as an ordered pair, e.g., "(1,5)". If an extreme value is attained twice, enter a comma-separated list of ordered pairs. If there are no absolute extrema of a given type, enter "none". D(x, y) = 5x²8xy+9y² +8 on the closed triangular region in the first quadrant bounded by the lines x = 0, y = 7, y = x. attained at (x, y) = attained at (x, y) = Absolute maximum value is Absolute minimum value is ⠀ ⠀ ⠀⠀
Find all absolute extrema of each function. Enter each point as an ordered pair, e.g., "(1,5)". If an extreme value is attained twice, enter a comma-separated list of ordered pairs. If there are no absolute extrema of a given type, enter "none". D(x, y) = 5x²8xy+9y² +8 on the closed triangular region in the first quadrant bounded by the lines x = 0, y = 7, y = x. attained at (x, y) = attained at (x, y) = Absolute maximum value is Absolute minimum value is ⠀ ⠀ ⠀⠀
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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Question
![### Finding Absolute Extrema of a Function
To determine the absolute extrema (maximum and minimum values) of each function, follow these steps:
1. **Identify the Region:**
- The given function is \( D(x, y) = 5x^2 - 8xy + 9y^2 + 8 \).
- The region of interest is a closed triangular area in the first quadrant, which is bounded by the lines:
- \( x = 0 \)
- \( y = 7 \)
- \( y = x \)
2. **Input the Extremum Values:**
- For absolute maximum and minimum values, insert each point as an ordered pair, e.g., "(1, 5)".
- If an extreme value is attained at more than one point, present them as a comma-separated list of ordered pairs.
- If there are no absolute extrema of a given type, enter "none".
#### Function and Boundary
The function is applied to the defined closed triangular region:
\[ D(x, y) = 5x^2 - 8xy + 9y^2 + 8 \]
#### Task
- **Absolute maximum value**: [Input Box], attained at \( (x, y) = \) [Input Box]
- **Absolute minimum value**: [Input Box], attained at \( (x, y) = \) [Input Box]
You must determine and input the coordinates where these extrema occur within the specified region. The boxes are provided for entering these values accordingly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4663b99b-8fe1-424a-98b9-10bd14779be6%2F30e42947-868e-42c2-b45e-e56523061f6a%2Fmgfzf33_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding Absolute Extrema of a Function
To determine the absolute extrema (maximum and minimum values) of each function, follow these steps:
1. **Identify the Region:**
- The given function is \( D(x, y) = 5x^2 - 8xy + 9y^2 + 8 \).
- The region of interest is a closed triangular area in the first quadrant, which is bounded by the lines:
- \( x = 0 \)
- \( y = 7 \)
- \( y = x \)
2. **Input the Extremum Values:**
- For absolute maximum and minimum values, insert each point as an ordered pair, e.g., "(1, 5)".
- If an extreme value is attained at more than one point, present them as a comma-separated list of ordered pairs.
- If there are no absolute extrema of a given type, enter "none".
#### Function and Boundary
The function is applied to the defined closed triangular region:
\[ D(x, y) = 5x^2 - 8xy + 9y^2 + 8 \]
#### Task
- **Absolute maximum value**: [Input Box], attained at \( (x, y) = \) [Input Box]
- **Absolute minimum value**: [Input Box], attained at \( (x, y) = \) [Input Box]
You must determine and input the coordinates where these extrema occur within the specified region. The boxes are provided for entering these values accordingly.
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