z=sin(xy); ∂z/∂x=ycos(xy); ∂z/∂y=xcos(xy); ∂2z/∂x2=-y2sin(xy); ∂2z/∂y2=-x2sin(xy); ∂2z/∂x∂y=∂2z/∂y∂x=-xysin(xy).
∂z/∂x=0=ycos(xy) when xy=(2n+1)π/2; and ∂z/∂y=0=xcos(xy) when xy=(2n+1)π/2 (where n is an integer).
z=sin(xy)=1 or -1 when xy=(2n+1)π/2.
The second derivative test D(x,y)=(-y2sin(xy))(-x2sin(xy))-(-xysin(xy))2=0. The nature of the critical points is inconclusive.
The actual surface of z is a set of continuous ridges (as in mountain ridges) forming a continuous rift (as valleys between the mountain ridges).
xy=(2n+1)π/2 is a set of rectangular hyperbolae, which forms a set of trajectories for mountain peaks and valleys.
So there are no local maxima and minima, just a set of ridges and valleys, infinitely long. The picture below shows the undulating surface. The hyperbolic shape formed by the mountains and valleys is evident in this representation.