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Find any relative(local) extrema and saddle points of f(x,y)=x+y^-4xy+1.

Find any relative(local) extrema and saddle points of f(x,y)=x+y^-4xy+1.

Advanced Engineering Mathematics
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 Problem: Finding Extrema and Saddle Points **Problem Statement:** 1. Find any relative (local) extrema and saddle points of the function \( f(x,y) = x^4 + y^4 - 4xy + 1 \). **Explanation:** To solve this problem, you will need to use techniques from multivariable calculus, specifically finding the first and second partial derivatives of the function and analyzing the critical points. **Steps:** 1. **Find the First Partial Derivatives:** - Calculate \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\). 2. **Find the Critical Points:** - Set \(\frac{\partial f}{\partial x} = 0\) and \(\frac{\partial f}{\partial y} = 0\), and solve for \(x\) and \(y\). 3. **Apply the Second Derivative Test:** - Compute the second partial derivatives: \(\frac{\partial^2 f}{\partial x^2}\), \(\frac{\partial^2 f}{\partial y^2}\), and \(\frac{\partial^2 f}{\partial x \partial y}\). - Use the second derivative test (determine the determinant of the Hessian matrix) to classify the nature of each critical point found in step 2. By following these steps, you can identify the relative extrema (local maxima and minima) and saddle points of the given function \(f(x,y) = x^4 + y^4 - 4xy + 1\).
Transcribed Image Text:### Calculus Problem: Finding Extrema and Saddle Points **Problem Statement:** 1. Find any relative (local) extrema and saddle points of the function \( f(x,y) = x^4 + y^4 - 4xy + 1 \). **Explanation:** To solve this problem, you will need to use techniques from multivariable calculus, specifically finding the first and second partial derivatives of the function and analyzing the critical points. **Steps:** 1. **Find the First Partial Derivatives:** - Calculate \(\frac{\partial f}{\partial x}\) and \(\frac{\partial f}{\partial y}\). 2. **Find the Critical Points:** - Set \(\frac{\partial f}{\partial x} = 0\) and \(\frac{\partial f}{\partial y} = 0\), and solve for \(x\) and \(y\). 3. **Apply the Second Derivative Test:** - Compute the second partial derivatives: \(\frac{\partial^2 f}{\partial x^2}\), \(\frac{\partial^2 f}{\partial y^2}\), and \(\frac{\partial^2 f}{\partial x \partial y}\). - Use the second derivative test (determine the determinant of the Hessian matrix) to classify the nature of each critical point found in step 2. By following these steps, you can identify the relative extrema (local maxima and minima) and saddle points of the given function \(f(x,y) = x^4 + y^4 - 4xy + 1\).
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