The function is
#f(x,y)=xy(1-8x-7y)=xy-8x^2y-7xy^2#
Caculate the partial derivatives
#(delf)/(delx)=y-16xy-7y^2#
#(delf)/(dely)=x-8x^2-14xy#
The critical points are
#{(y-16xy-7y^2=0),(x-8x^2-14xy=0):}#
#<=>#, #{(y(1-16x-7y)=0),(x(1-8x-14y)=0):}#
Therefore, #(0,0)# is a point
#<=>#, #{((16x+7y)=1),((8x+14y)=1):}#
#<=>#, #{((16x+7y)=1),((16x+28y)=2):}#
#<=>#, #{(16x+7y=1),(y=1/21):}#
#<=>#, #{(x=1/24),(y=1/21):}#
The other point is #(1/24, 1/21)#
Calculate the second derivatives
#(del^2f)/(delx^2)=-16y#
#(del^2f)/(dely^2)=-14x#
#(del^2f)/(delxdely)=1-16x-14y#
#(del^2f)/(delydelx)=1-16x-14y#
Calculate the Determinant #D(x,y)# of the hessian Matrix
#((-16y,1-16x-14y ),(1-16x-14y,-14y))#
#D(x,y)=224y^2-(1-16x-14y)^2#
Therefore,
#D(0,0)=-1#
As #D(0,0)<0#, this is a saddle point.
#D(1/24,1/21)=0.51-0.11=0.4#
#D(1/24,1/21)>0#, then #(del^2f(1/24,1/21))/(delx^2)=-16/21#
#(del^2f(1/24,1/21))/(delx^2)<0#
This is a local maximum at #(1/24,1/21)#
