Local min, max or relative min,max?

The answer to your second question depends on the precise definition of relative extremum being used.

If the definition of relative minimum (for example) being used is something like:

Definition 1
#f(c)# is a relative (local) minimum if and only if there is an open interval #(a,b)# containing #c# for which, for all #x# in #(a,b)#, we have #f(x) >= f(c)#

Then the value at an endpoint of a domain is not a relative (local) minimum.

#f(x) = sqrtx# has no relative minimum.
#g(x) = arcsin(x)# has no relative minimum
(This is the definition used in James Stewart's Calculus and appears to be that used by WolframAlpha.)

Definition 2
If the definition adds the restriction "In the domain of #f#" as follows:

#f(c)# is a relative (local) minimum if and only if there is an open interval #(a,b)# containing #c# for which, for all #x# in the domain of #f# and in #(a,b)#, we have #f(x) >= f(c)#.

Then the value at an endpoint of a domain could be a relative (local) minimum.

#f(x) = sqrtx# has relative minimum #0# at #x=0#.
#g(x) = arcsin(x)# has relative minimum #-pi/2# at #x=-1#.