Find the critical points of f (x, y) = 8y4 + x² + xy - 3y² - y³. Use the contour map to determine their nature (local minimum, local maximum, and saddle point). of al 0.2 0.3 -0.3 -0.2 -0.1 local minimum: 6.1 6.2 Use symbolic notation and fractions where needed. Enter DNE if the answer does not exist. List the points the comma if there are more than one.)
Find the critical points of f (x, y) = 8y4 + x² + xy - 3y² - y³. Use the contour map to determine their nature (local minimum, local maximum, and saddle point). of al 0.2 0.3 -0.3 -0.2 -0.1 local minimum: 6.1 6.2 Use symbolic notation and fractions where needed. Enter DNE if the answer does not exist. List the points the comma if there are more than one.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Problem 1RQ
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![### Finding Critical Points of a Given Function Using a Contour Map
To enhance your understanding of finding critical points, we will analyze the function \( f(x, y) = 8y^4 + x^2 + xy - 3y^2 - y^3 \) using its contour map to determine the nature of the critical points (local minimum, local maximum, and saddle points).
#### Function
\[ f(x, y) = 8y^4 + x^2 + xy - 3y^2 - y^3 \]
#### Instructions
1. **Analyze the Contour Map**: Use the provided contour map below to determine the nature of the critical points for the function.
2. **Categorize the Points**: Based on the contour map, categorize each critical point as a local minimum, local maximum, or a saddle point.
3. **Use Symbolic Notation**: Use symbolic notation and fractions where necessary.
4. **Checklist**:
- Enter **DNE** if the answer does not exist.
- List multiple points separated by commas.
#### Contour Map
- The contour map has labeled points where the function's value remains constant.
- The contour lines are marked with values such as 0.1, 0.2, 0.3, and so on.
- Examine how these lines change around the points to identify the nature of the critical points.
#### Categories
1. **Local Minimum**:
- Represented by a valley in the contour plot, where the function value is lower than all nearby points.
- Filled box for response:
2. **Local Maximum**:
- Represented by a peak in the contour plot, where the function value is higher than all nearby points.
- Filled box for response:
3. **Saddle Points**:
- Represent places where the function switches direction (i.e., changes from increasing to decreasing or vice versa).
- Filled box for response:
#### Critical Points Identification
- Fill in the boxes below with the identified critical points. If there are multiple points, separate them with a comma.
##### (Use symbolic notation and fractions where needed. Enter DNE if the answer does not exist. List the points separated by the comma if there are more than one.)
- **Local Minimum**:
- **Input Box**:
- **Local Maximum**:
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F463f1f77-6106-4bd7-a4ec-d9f0e00615dd%2F0397b99f-9632-4475-b8f6-793ed599c49a%2Femnik4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding Critical Points of a Given Function Using a Contour Map
To enhance your understanding of finding critical points, we will analyze the function \( f(x, y) = 8y^4 + x^2 + xy - 3y^2 - y^3 \) using its contour map to determine the nature of the critical points (local minimum, local maximum, and saddle points).
#### Function
\[ f(x, y) = 8y^4 + x^2 + xy - 3y^2 - y^3 \]
#### Instructions
1. **Analyze the Contour Map**: Use the provided contour map below to determine the nature of the critical points for the function.
2. **Categorize the Points**: Based on the contour map, categorize each critical point as a local minimum, local maximum, or a saddle point.
3. **Use Symbolic Notation**: Use symbolic notation and fractions where necessary.
4. **Checklist**:
- Enter **DNE** if the answer does not exist.
- List multiple points separated by commas.
#### Contour Map
- The contour map has labeled points where the function's value remains constant.
- The contour lines are marked with values such as 0.1, 0.2, 0.3, and so on.
- Examine how these lines change around the points to identify the nature of the critical points.
#### Categories
1. **Local Minimum**:
- Represented by a valley in the contour plot, where the function value is lower than all nearby points.
- Filled box for response:
2. **Local Maximum**:
- Represented by a peak in the contour plot, where the function value is higher than all nearby points.
- Filled box for response:
3. **Saddle Points**:
- Represent places where the function switches direction (i.e., changes from increasing to decreasing or vice versa).
- Filled box for response:
#### Critical Points Identification
- Fill in the boxes below with the identified critical points. If there are multiple points, separate them with a comma.
##### (Use symbolic notation and fractions where needed. Enter DNE if the answer does not exist. List the points separated by the comma if there are more than one.)
- **Local Minimum**:
- **Input Box**:
- **Local Maximum**:
-
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