Multiple Variable Min Max Question

[stau40]

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?

 

[Dick]

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?

 

[stau40]

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.

 

[Dick]

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.