Critical points, open and closed interval I know that end-points on a closed interval $[a,b]$ are critical points, but what about end points on an open interval $(a,b)$? 
I remember my teacher saying that they are also considered to critical points but since $a$ and $b$ are not in the domain of the function how can this be true?
 A: By definition a point $x_0$ is a critical point of $f$ if $f$ is defined in some open neighborhood of $x_0$, and $f'(x_0)=0$.
Faced with an extremal problem for a continuous function $f:\>[a,b]\to{\mathbb R}$ you set up a candidate list  consisting of (i) the critical points of $f$ in $\ ]a,b[\ $, (ii) the points $a$ and $b$, and (iii) the points in $\ ]a,b[\ $ where $f$ is not differentiable. If the resulting list $S$ is finite, i.e., $S=\{x_1,x_2,\ldots, x_m\}$ for some $m\in{\mathbb N}$, then
$$\max_{a\leq x\leq b}f(x)=\max_{1\leq i\leq m}f(x_i)\ ,$$
and similarly for the minimum. Note that we don't need to compute second derivatives in this process.
A: The maximum or minimum values of a function can occur at the endpoints $a$ and $b$ or the interior $(a,b)$, obviously. If $x\in (a,b)$ is a critical point, then the derivative $f'(x)=0$ or $f'(x)$ is undefined. It is these $x$ values, that we call critical points.
A: Your teacher probably wants you to consider the endpoints as critical points because on a closed interval like that, a function may take a maximum or minimum value at those endpoints. I've seen that the definition of "critical point" can vary in different calculus texts.
A: I'd guess that this person's teacher didn't communicate clearly.  a and b are not critical numbers if the interval is open.  But for a problem where you need to find an absolute maximum or minimum, the endpoints still need to be considered even if they're not in the domain, because they could show that the function has no max or min on the open interval.  
For example, f(x)=x^2 has its [-1, 5] at the critical number x=5.  But f(x)=x^2 has no maximum on (-1, 5) because of the open endpoints.  If you didn't bother to check the endpoints because they're not critical points, then you might not realize the lack of maximum value.  I expect the teacher meant for you to check the endpoints in a problem such as this even though they are not critical points, and just misspoke.
