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math about to have a test over this please help!

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Unfortunately that is the entire problem..
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! - ,

    F(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

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