Critical points and 2nd partials test

[camino]

Homework Statement

Find all critical points of f(x,y) = 12xy-x^2 y-2xy^2 and test them for relative extrema using the 2nd partials test.

Homework Equations

The Attempt at a Solution

{0=fx(x,y)=12-2x-2 10x-2 x=5
{0=fy(x,y)=12-4y y=3

Critical point: (5,3)

fxx(x,y)= -2
fyy(x,y)= -4
fxy(x,y)= 0

d(5,3)=fxx(5,3)fyy(5,3)-(fxy(5,3))^2
d=(0)(0)-(0)
d=0 no conclusion
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Just wondering if I did this right?

 

[HallsofIvy]

Homework Statement

Find all critical points of f(x,y) = 12xy-x^2 y-2xy^2 and test them for relative extrema using the 2nd partials test.

Homework Equations

The Attempt at a Solution

{0=fx(x,y)=12-2x-2 10x-2 x=5
{0=fy(x,y)=12-4y y=3

Critical point: (5,3)

fxx(x,y)= -2
fyy(x,y)= -4
fxy(x,y)= 0

d(5,3)=fxx(5,3)fyy(5,3)-(fxy(5,3))^2
d=(0)(0)-(0)
d=0 no conclusion
---------------------------------------

Just wondering if I did this right?

fxx(5,3)fyy(5,3)= (-2)(-4), not (0)(0)!

 

[camino]

Ok, so

d=(-2)(-4)-(0)
d=8

Is that right?