Question

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?

Answer 1

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`.