In my life, I have been considering the definition of absolute maximum as "the highest point of a graph", if it does not diverge to infinity, of course.
However, I attended a lesson in which it was taught that "absolute maximum is the highest of the relative maxima".
In other words, they're defining global maximum as the highest local maximum.
This can be true for functions defined in the whole real domain. Nevertheless, if we constrain the domain to a certain closed interval [a,b], it can be that f(a) or f(b) is actually higher than a possible local maximum located inside the interval. For example, f(x)=x^3-2x;\qquad x\in[-1,2].
According to the first definition, (2, f(2)) is an absolute maximum, but according to the second one, it would not be. On the contrary, the absolute maximum would be located on the local maxium, as it is the only one in the graph.
The questions are: which one of the two definitions is true? Are those two definitions coexisting, forming two streams? Which books can support each theory? Thanks in advance.
