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Multiple Variable Min Max Question

  • Thread starter stau40
  • Start date
  • #1
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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?
 

Answers and Replies

  • #2
Dick
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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?
 
  • #3
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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.
 
  • #4
Dick
Science Advisor
Homework Helper
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618
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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