● Question 6 ▼ < > Apply a second derivative to identify a critical points as a local maximum, local minimum or saddle point for a function. Let f(x, y) = x² + 2y¹ + 24xy. List the critical points: Submit Question The critical point at the origin is a Select an answer The critical point with positive a value is a Select an answer ✓ Select an answer Question Help: Video maximum minimum saddle point
● Question 6 ▼ < > Apply a second derivative to identify a critical points as a local maximum, local minimum or saddle point for a function. Let f(x, y) = x² + 2y¹ + 24xy. List the critical points: Submit Question The critical point at the origin is a Select an answer The critical point with positive a value is a Select an answer ✓ Select an answer Question Help: Video maximum minimum saddle point
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
![**Title: Analyzing Critical Points Using the Second Derivative**
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**Objective:**
Apply the second derivative to identify critical points as a local maximum, local minimum, or saddle point for a function.
---
**Problem Statement:**
Let \( f(x, y) = x^2 + 2y^4 + 24xy \).
1. **List the critical points:**
- [Input Box]
2. **Determine the type of critical points:**
- The critical point at the origin is a [Dropdown Menu: Select an answer].
- The critical point with positive \( x \) value is a [Dropdown Menu: Select an answer].
- Options: maximum, minimum, saddle point
---
**Additional Resources:**
**Question Help:** [Video Icon]
[Button: Submit Question]
---
**Instructions for Students:**
To solve this problem, use the second partial derivatives to construct the Hessian matrix. Evaluate the determinant and trace of the Hessian to classify each critical point:
1. Calculate \(\frac{\partial f}{\partial x}\), \(\frac{\partial f}{\partial y}\), and set them to zero to find critical points.
2. Use the second derivatives \(\frac{\partial^2 f}{\partial x^2}\), \(\frac{\partial^2 f}{\partial y^2}\), and \(\frac{\partial^2 f}{\partial x \partial y}\).
3. Compute the determinant of the Hessian matrix: \(\text{Det}(H) = \left( \frac{\partial^2 f}{\partial x^2} \right)\left( \frac{\partial^2 f}{\partial y^2} \right) - \left( \frac{\partial^2 f}{\partial x \partial y} \right)^2\).
4. Use the determinant and the trace to categorize each critical point as either a local maximum, local minimum, or a saddle point.
Please watch the provided help video for a step-by-step process on how to compute the second derivative and analyze the critical points effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c62a616-6a2f-456f-ac81-c6090d5022b3%2Fe6b63093-f84f-4f7e-a303-b3264d0af94c%2Fw9yk01l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Analyzing Critical Points Using the Second Derivative**
---
**Objective:**
Apply the second derivative to identify critical points as a local maximum, local minimum, or saddle point for a function.
---
**Problem Statement:**
Let \( f(x, y) = x^2 + 2y^4 + 24xy \).
1. **List the critical points:**
- [Input Box]
2. **Determine the type of critical points:**
- The critical point at the origin is a [Dropdown Menu: Select an answer].
- The critical point with positive \( x \) value is a [Dropdown Menu: Select an answer].
- Options: maximum, minimum, saddle point
---
**Additional Resources:**
**Question Help:** [Video Icon]
[Button: Submit Question]
---
**Instructions for Students:**
To solve this problem, use the second partial derivatives to construct the Hessian matrix. Evaluate the determinant and trace of the Hessian to classify each critical point:
1. Calculate \(\frac{\partial f}{\partial x}\), \(\frac{\partial f}{\partial y}\), and set them to zero to find critical points.
2. Use the second derivatives \(\frac{\partial^2 f}{\partial x^2}\), \(\frac{\partial^2 f}{\partial y^2}\), and \(\frac{\partial^2 f}{\partial x \partial y}\).
3. Compute the determinant of the Hessian matrix: \(\text{Det}(H) = \left( \frac{\partial^2 f}{\partial x^2} \right)\left( \frac{\partial^2 f}{\partial y^2} \right) - \left( \frac{\partial^2 f}{\partial x \partial y} \right)^2\).
4. Use the determinant and the trace to categorize each critical point as either a local maximum, local minimum, or a saddle point.
Please watch the provided help video for a step-by-step process on how to compute the second derivative and analyze the critical points effectively.
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