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itical points of the f(x, y) = x³-9xy + y³ - 2 maller y-value tical point ssification ger y-value Cical point ssification - (1 (x, y) = (x, y) = -Select- -Select- men use test to
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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![**Finding Critical Points and Classifying Their Nature**
To analyze the function \( f(x, y) = x^3 - 9xy + y^3 - 2 \), follow the steps below:
### Step 1: Find Critical Points
1. Identify the points where the gradient of the function is zero.
2. List these critical points and classify them based on the smaller and larger \( y \)-values.
- **Smaller \( y \)-value:**
- Critical Point \((x, y) = \) [Input Box for coordinates]
- Classification: [Dropdown menu options: select either saddle, minimum, maximum]
- **Larger \( y \)-value:**
- Critical Point \((x, y) = \) [Input Box for coordinates]
- Classification: [Dropdown menu options: select either saddle, minimum, maximum]
### Step 2: Use the Second Derivative Test
- Check the nature of each critical point using the second derivative test to determine if it is a local minimum, maximum, or saddle point. If this test fails to classify the point, classification might be unclear and require further investigation.
### Step 3: Determine Relative Extrema
- **Relative Minimum Value:** [Input Box]
- **Relative Maximum Value:** [Input Box]
*Note: If a relative extremum does not exist, enter DNE (Does Not Exist).*
This process will help you fully analyze the critical points and classify them using the tools of calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ce61022-bd56-4e8d-866e-9fc156b54e62%2F64836311-f755-44a1-9bc7-5ea34080b5b7%2Fmnvph9b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding Critical Points and Classifying Their Nature**
To analyze the function \( f(x, y) = x^3 - 9xy + y^3 - 2 \), follow the steps below:
### Step 1: Find Critical Points
1. Identify the points where the gradient of the function is zero.
2. List these critical points and classify them based on the smaller and larger \( y \)-values.
- **Smaller \( y \)-value:**
- Critical Point \((x, y) = \) [Input Box for coordinates]
- Classification: [Dropdown menu options: select either saddle, minimum, maximum]
- **Larger \( y \)-value:**
- Critical Point \((x, y) = \) [Input Box for coordinates]
- Classification: [Dropdown menu options: select either saddle, minimum, maximum]
### Step 2: Use the Second Derivative Test
- Check the nature of each critical point using the second derivative test to determine if it is a local minimum, maximum, or saddle point. If this test fails to classify the point, classification might be unclear and require further investigation.
### Step 3: Determine Relative Extrema
- **Relative Minimum Value:** [Input Box]
- **Relative Maximum Value:** [Input Box]
*Note: If a relative extremum does not exist, enter DNE (Does Not Exist).*
This process will help you fully analyze the critical points and classify them using the tools of calculus.
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