I would use either Extract Values to Points or Zonal Statistics as Table (same as zonal statistic, but with a table output). However, the former tends to not collect values for all points if there is a large number of points.
Using Zonal Statistic as Table I would do the following:
First, add a field to your point feature class, call it ID, and use field calculator to copy the OBJECTID (or FID) values into the ID field. The reason being that you need a unique value for the zonal statistic ZONE field.
Then with Zonal Statistic
- input raster or feature zone data -> point feature class (or shapefile)
- zone field -> the ID field mentioned above
- input value raster -> the DEM
- Output table -> \path\to\outputtable
- statistic type -> min or max (doesn't matter since there will be only 1 value)
Finally, join the output table to your feature class based on ID field.