I'm very confused with all the different definitions and conventions regarding how we should treat endpoints.
One book says the following: let $f(x)$ be a function defined on a segment $[a,b]$. If the left-derivative of $b$ is greater than zero then $b$ is called endpoint maximum (if it's less than zero – endpoint minimum). The same goes for $a$, but there we shall check the right-derivative.
The second book doesn't even allow the possibility of endpoint extrema. It just says that if any endpoint is the highest/lowest of all the points in the segment then they are simply "highest/lowest" points.
The third book also doesn't allow the possibility of endpoint extrema because it's not really a differentiable point (the derivative must exist at both sides).
The fourth book says that any endpoint is automatically a local extremum (it may be the highest/lowest point and then it will be called a global extremum).
Wolfram Alpha says that contrary to what the first book has defined, endpoint is an extremum only if it is the highest/lowest point of all. Say, for example, we have the function $f(x)=(x-1)^2$ for $x \in [0,3]$. Then, according to Wolfram, there is only one minimum (both global and local) which is $x=1$. There's also a global maximum (not clear whether it is also a local one) at $x=3$. But $x=0$ is not an extremum – not even a local maximum of the function, despite the fact that it has a negative right-derivative.
So why $x=0$ is not an extremum here? Does it imply that endpoint extremum can exist only of it is the highest/lowest point of all the points in the segment?
Best Answer
Great question.
First note that to some degree this is about conventions and not mathematical correctness, so there is not necessarily one right answer to the query.
That said, I think some points should be clarified. (I will use your example throughout)
With respect to the interval $[0,3]$ both endpoints must be local extrema. With respect to the entire domain the of the function ($\mathbb{R}$ in this case) both endpoints are not necessarily extrema, because they may or may not be the largest/smallest values of the function over the interval $[0,3]$. So it depends on what you are taking to be the "domain" of the function. In this sense the first and fourth books basically agree.
I'm not entirely sure what books 2 and 3 are getting at by "now allowing endpoint extrema". Whether or not you say the function is differentiable at the endpoints, as mentioned above, depends on whether you are restricting the domain of the function to the interval or not. But either way, the endpoints can certainly have the highest or lowest values of the function over the interval.
So in general: differentiate and set equal to $0$ to find "critical points". Then determine what local extrema lie inside the interval. Then check boundary points (in this case the two endpoints of the interval, in multi-variable calculus you will have to use Lagrange multipliers or a similar technique). Compare internal critical points and boundary points to find the absolute extrema (max and min) over the interval.
I think in general it is most consistent with mathematical convention to refer to the points of extrema as "extrema", whether or not they are at the endpoints.