So my question is:
- Is a global extremum necessarily a also local extremum?
I think the answer is no if you define that f has a local max at c if f(c) ≥ f(x) for all x near c, even for those x not in the domain, then an endpoint which is a global maximum of the domain is not a local maximum since we can find points in a neighborhood of c such that f(c) < f(x) even if that x is not in the domain. It would be also local if we defined local max as f(c) ≥ f(x) for all x of the domain near c.
- Can a stationary point be something OTHER than a local extrema or saddle point?
I had the 2) on an exam and the correct answer was TRUE. the above definition for local extremum explains why we could have a extremum on an endpoint that is global but not local. But here I think the trick is with the definition of stationary point. In my class we distinguish stationary points (where the gradient is 0) and critical points (where the gradient does not exist.) My reasoning is this: A critical point can be a point where the gradiant is not defined i.e an endpoint. And an endpoint can be an absolute extrema but cannot be a local extrema. Thus the statement is true because a stationary point also could be global extremum rather than a local extremum or saddle point.
I had a discussion with a friend:
He told me that a stationary point is not necessarily a point of the function that is differentiable.
I thought that a to be a stationary point that point must cancel the gradient. But he told me that that was true only if we know that the function is differentiable at that point.
And a teacher told him that
if it is not mentioned that f is differentiable then a stationary point is not always a local min, local max or saddle.
That really confused me because I thought stationary point where only defined on functions where the gradient exists and is equal to 0 at that point.
EDIT: From wikipedia: "If the function is smooth, or, at least twice continuously differentiable, a critical point may be either a local maximum, a local minimum or a saddle point." Does this imply that the function being smooth is a necessity thus implying that a critical point can be other than an extremum or saddle point? So what happens if the function is not smooth?