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April 29, 2016
Posted by **Lucy** on Tuesday, February 28, 2012 at 4:51pm.

Find and classify all local minima, local max, and saddle points of the function F(x,y)= -3x^2y-3xy^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!!

- math about to have a test over this please help! -
**MathMate**, Tuesday, February 28, 2012 at 9:21pmF(x,y)=-3x^2y-3xy^2+36xy

**For maximum/minimum,Fx=0 AND Fy=0**

Fx=36y-6xy-3y^2=3y(12-2x-y)=0 ...(1)

Fy=36x-6xy-3x^2=3x(12-2y-x)=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

12-2x-y=0 and

12-x-2y=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(6-x-y) [Clairot's theorme]

Now calculate

D(x,y)=Fxx.Fyy-Fxy²

=36xy-6(36-12(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