What are the extrema and saddle points of #f(x, y) = xy(1-x-y)#?

Question

What are the extrema and saddle points of #f(x, y) =  xy(1-x-y)#?

Isabella Ackerman · Accepted Answer

The points #(0,0),(1,0)#, and #(0,1)# are saddle points. The point #(1/3,1/3)# is a local maximum point.

We can expand #f# to #f(x,y)=xy-x^2y-xy^2#. Next, find the partial derivatives and set them equal to zero.

#\frac{\partial f}{\partial x}=y-2xy-y^2=y(1-2x-y)=0#

#\frac{\partial f}{\partial y}=x-x^2-2xy=x(1-x-2y)=0#

Clearly, #(x,y)=(0,0),(1,0),# and #(0,1)# are solutions to this system, and so are critical points of #f#. The other solution can be found from the system #1-2x-y=0#, #1-x-2y=0#. Solving the first equation for #y# in terms of #x# gives #y=1-2x#, which can be plugged into the second equation to get #1-x-2(1-2x)=0 => -1+3x=0 =>x=1/3#. From this, #y=1-2(1/3)=1-2/3=1/3# as well.

To test the nature of these critical points, we find second derivatives:

#\frac{\partial^{2}f}{\partial x^{2}}=-2y#, #\frac{\partial^{2}f}{\partial y^{2}}=-2x#, and #\frac{\partial^{2}f}{\partial x\partial y}=\frac{\partial^{2}f}{\partial y\partial x}=1-2x-2y#.

The discriminant is therefore:

#D=4xy-(1-2x-2y)^2#

#=4xy-(1-2x-2y-2x+4x^2+4xy-2y+4xy+4y^2)#

#=4x+4y-4x^2-4y^2-4xy-1#

Plugging the first three critical points in gives:

#D(0,0)=-1<0#, #D(1,0)=4-4-1=-1<0#, and #D(0,1)=4-4-1=-1<0#, making these points saddle points.

Plugging in the last critical point gives #D(1/3,1/3)=4/3+4/3-4/9-4/9-4/9-1=1/3>0#. Also note that #\frac{\partial^{2}f}{\partial x^{2}}(1/3,1/3)=-2/3<0#. Therefore, #(1/3,1/3)# is a location of a local maximum value of #f#. You can check that the local maximum value itself is #f(1/3,1/3)=1/27#.

Below is a picture of the contour map (of level curves) of #f# (the curves where the output of #f# is constant), along with the 4 critical points of #f#.

Samantha Adkison · Answer

undefined To find the critical points, solve ∇f(x, y) = 0. Differentiate f(x, y) with respect to x and y, then set both equal to zero. You'll get two critical points: (0, 0) and (1, 0). To determine their nature, evaluate the Hessian matrix at each critical point. The Hessian matrix for f(x, y) is [[2y - 6xy, 2x - 3x^2 - y], [2x - 3y - y^2, 2y - 3xy]]. Evaluate the determinant of the Hessian at each critical point. If it's positive, it's a minimum; if negative, a maximum; if zero, a saddle point. At (0, 0), the determinant is negative, so it's a saddle point. At (1, 0), the determinant is positive, so it's a minimum.