Going a little old school
Start at origin (x,y,z)=(0,0,0)
On the surface
planar projection on y axis (y=0)
4x^2=16z Parabola opening upward
planar projection on x axis (x=0)
y^2=16z Parabola opening upward
so surface point on origin (0,0,0) is a minimum.
Now do some horizontal projections
Oval minor axis 2, major axis 4
Oval minor axis 4, major axis 8
Oval minor axis 8, major axis 16
Is an oval shaped paraboloid opening upward from the origin.