Question

Find and classify all local minima, local maxima, and saddle points of the function f(x,y)= -3yx^2-3xy^2+36xy

Answer 1

F = -3yx^2-3xy^2+36xy
Fx = -6xy - 3y^2 + 36y
Fxx = -6y

Fy = -3x^2 - 6xy + 36x
Fyy = -6x

Fxy = -6x - 6y + 36

D = FxxFyy-(Fxy)^2 = 36xy - 36(x+y-6)^2

Fx = 0 Fy=0 at (0,0)
D<0 so a saddle point


Fx = -6xy - 3y^2 + 36y
Fx = 0 when y = 2(6-x)

Fy = -3x^2 - 6xy + 36x
Fy = 0 when x = 2(6-y)

So there is a local max for z along those two lines

Answer 2

Okay Im just confused as how to get Fx=0 and Fy=0, If I set -6xy-3y^2+36y=0 how do i solve this?? Algebra was so long ago!
could you detail that part of the problem...