Multiple Variable Min Max Question

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Homework Help Overview

The discussion revolves around finding the critical points of the function z = (x^5)y + (xy^5) + xy, which involves multiple variables. The subject area includes calculus, specifically the analysis of critical points in multivariable functions.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the process of finding critical points by setting the partial derivatives fx and fy to zero. There is an exploration of the implications of factoring the equations and questioning whether x = 0 and y = 0 is the only solution.

Discussion Status

The discussion is active, with participants providing guidance on factoring the equations and questioning the validity of the conclusions drawn about the critical points. There is acknowledgment that the second factors cannot be zero, which leads to further exploration of the implications.

Contextual Notes

Participants are working under the constraints of the problem statement and are examining the assumptions regarding the nature of the critical points derived from the equations.

stau40
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Homework Statement


Find the critical point for z=(x^5)y+(xy^5)+xy


Homework Equations


fx(x,y)=(5x^4)y+(y^5)+y=0
fy(x,y)=x^5+(5xy^4)+x=0

The Attempt at a Solution


After finding fx and fy shown above, I attempt to find the critical points in one of the equations above, but the only number that works (that I can think of) is x=0 or y=0 and this doesn't seem correct to me after substituting it into the other equation. Am I doing something wrong?
 
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You aren't doing anything wrong. Factor a y out of the first equation and an x out of the second. Can you make an argument that x=y=0 is the only critical point?
 
After factoring I'm left with:

fx(x,y)=y(5x^4+y^4+1)=0
fy(x,y)=x(x^4+5y^4+1)=0

Since both equations are to an even power (can't be negative), it brings me to the conclusion that only x=y=0 works.
 
stau40 said:
After factoring I'm left with:

fx(x,y)=y(5x^4+y^4+1)=0
fy(x,y)=x(x^4+5y^4+1)=0

Since both equations are to an even power (can't be negative), it brings me to the conclusion that only x=y=0 works.

Right. The second factors can never be zero. So the first must.
 

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