Question

Can the endpoints of a function be a local/absolute maximum/minimum?

Answer 1

Consider the function \[f(x) = \sqrt{x}\]

Or an arbitrary function of your choice. Any point can be a local min or max, or a global min or max.

Answer 2

After some research I will retract what I said. Endpoints can be global but not local because of the way local extrema is defined. There would have to be a neighborhood around a point in order for it to be considered a local minimum or maximum.

So end points can be global extrema but not local ones.