z=2x+y and y=x²⇒z=x²+2x.
dz/dx=2x+2=0 at an extremum, so solving for x, x=-1. m=dz/dx increases as x increases from -1. (When x<-1, m<0. At (-2,4,0), m=-2, |m|=2. At (2,4,8), |m|=m=6.)
Therefore the point (-1,1,-1) is an extremum. This is a minimum because the starting point (-2,4,0) is in the x-y plane and (-1,1,-1) is below this plane. Also, the crawl path passes through the origin (0,0,0). The bug’s path from (-2,4,0) to (2,4,8) dips below the x-y plane to the minimum point (-1,1,-1), rises through the origin, and proceeds along the curve z=x²+2x. In the 3D picture (best viewed using 3D glasses, right eye green or blue, left eye red), points on the parabola z=x²+2x are emphasised by the labelled “beads” strung along the crawl path, where A is the start and B the finish. Bead K is the minimum. The projection of the parabola z=x²+2x on to the x-y plane is y=x², and is shown as an upright extrusion of the parabola y=x². Effectively, the plane z=2x+y slices through the extruded parabola to produce the beaded path.
|m|=0 is minimum and |m|=6 is maximum.