Local min, max or relative min,max?

What's the difference between a local max, min and a relative max min? Also, they can be cusps such as corner and kincks, but not endpoints right?

1 Answer
Apr 21, 2017

I have always (40 years) seen local and relative used to mean exactly the same thing when applied to extrema.

Explanation:

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.