This is a question of convention - specifically the convention used in Stewart's Calculus.
In Stewart's calculus (the latest version), in chapter 4.1, definition 6 defines a critical number of a function $f$ to be a number $c$ in the domain of $f$ such that either $f'(c) = 0$ or $f'(c)$ does not exist.
Suppose $f$ is obtained from a nice function by restricting to a closed interval $[a,b]$. Say, $g(x) = x$ and $f = g|_{[0,1]}$. Does Stewart consider 0 and 1 to be critical points of $f$? In other words, does $f'(0)$ and $f'(1)$ exist?
The book seems to be extremely elusive above this, going so far as to not including any exercises that might elucidate which convention he uses.
I'm asking this because I'm an instructor trying to decide what convention to take. I'm hoping that by choosing one convention over the other, I do not inadvertently subtly contradict Stewart somewhere down the road.
In Stewart's description of "The closed interval method" (Section 4.1), he first asks you to find the values of $f$ at critical numbers in $(a,b)$, and then to compute the values of $f$ at the endpoints $a$ and $b$ (and doesn't simply say compute values of $f$ at all critical numbers). The fact that he does this would suggest that he does not consider the endpoints to be critical points. On the other hand, taking this convention would seem to imply that $f'(a)$ does exist, but he defines $f'(a) := \lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}$, and he only ever defines the limit of a function at points on the interior of its domain, so technically according to Stewart this limit is "undefined", so "does not exist", but then it is very puzzling why he phrases the closed interval method in the way that he does.
You are correct. The derivative does not exist at the endpoints based on Stewart's definition, and therefore by Stewarts' definition endpoints are critical points. However, as Ninad Munshi, suggests, do not focus on this. If a student asks, you can have the same discussion we are having here. Whether you test endpoints to find a global max or min or you include them as "critical points" is only a matter of naming.