math about to have a test over this please help!
posted by Lucy .
Unfortunately that is the entire problem..
Find and classify all local minima, local max, and saddle points of the function F(x,y)= 3x^2y3xy^2+36xy
There should be 3 saddle points and a maxima for sure! Please help me get the points. I can't do these maxima minima problems. they are confusing as heck! Im about to have a test over this, freaking out!!

F(x,y)=3x^2y3xy^2+36xy
For maximum/minimum,Fx=0 AND Fy=0
Fx=36y6xy3y^2=3y(122xy)=0 ...(1)
Fy=36x6xy3x^2=3x(122yx)=0 ...(2)
Clearly (0,0) is a critical point.
If y=0, solve for x in (2) to get x=12, so
(12,0) is another critical point.
Similarly (0,12) is another critical point.
Finally, solve for x,y in
122xy=0 and
12x2y=0 to get
x=y=4, or (4,4) is a critical point.
So the four critical points are
(0,0)
(0,12)
(12,0)
(4,4)
Second derivative test
Calculate
Fxx=6y
Fyy=6x
Fxy=Fyx=6(6xy) [Clairot's theorme]
Now calculate
D(x,y)=Fxx.FyyFxy²
=36xy6(3612(x+y)+(x+y)^2)
D(0,0)=216<0 => saddle point
D(0,12)=216<0 => saddle point
D(12,0)=216<0 => saddle point
D(4,4)=552 >0 => max or min
Fxx=6(4)=24 <0 => maximum