(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find all solutions (x,y) for which fx(x,y) = 0 = fy(x,y) if f(x,y) = 12xy - x^2 y - 2xy^2

2. Relevant equations

3. The attempt at a solution

f(x,y)=12xy-x^2y-2xy^2

fx(x,y)=12y-2xy-2y^2

fy(x,y)=12x-x^2-4xy

0=12y-2xy-2y^2

0=12x-x^2-4xy

EQ 1: 2xy=12y-2y^2

2x=12-2y

x=6-y

EQ 2: 0=12(6-y)-(6-y)^2-4(6-y)y

0=72-12y-(y^2-12y+36)-24y+4y^2

0=3y^2-24y+36

0=3(y^2-8y+12)

0=3(y-6)(y-2)

y=6 y=2

x=0 x=4 so (0,6) , (4,2)

I found those 2 solutions so far, is there any more that I might have missed?

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# Finding partial derivatives

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