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Find the relative maximum and minimum values for: f(x,y)=x² + y³ - 3xy.

Find the relative maximum and minimum values for: f(x,y)=x² + y³ - 3xy.

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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**Problem Statement:** Find the relative maximum and minimum values for the function: \[ f(x, y) = x^3 + y^3 - 3xy \] The task involves finding the critical points of the given function \( f(x, y) \) and determining whether these points are relative maxima, minima, or saddle points using appropriate mathematical techniques, such as the second partial derivative test.
Transcribed Image Text:**Problem Statement:** Find the relative maximum and minimum values for the function: \[ f(x, y) = x^3 + y^3 - 3xy \] The task involves finding the critical points of the given function \( f(x, y) \) and determining whether these points are relative maxima, minima, or saddle points using appropriate mathematical techniques, such as the second partial derivative test.
Expert Solution
Step 1: Definition

Let f(x,y) be a real valued function, then the critical points occur at fx=0,fy=0. 

Let x* be a critical point .

  • If fxx(x∗)>0,D=fxxfyy−fxy2>0 , then x* is a relarive minimum .
  • If fxx(x∗)<0,D=fxxfyy−fxy2>0 , then x* is a relative miximum .
  • if D=fxxfyy−fxy2<0 , then x* is a saddle point .
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