0
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ The endpoint of the domain can be a local maximum (with no condition on the derivative). If it is, you must consider it when finding the largest local maximum. $\endgroup$ Feb 7, 2021 at 1:45

1 Answer 1

0
$\begingroup$

The absolute maximum is a value $M$ in the image such that $f(x) \leq M$ for all $x$ in the domain. So, the first definition is accurate here and reflects how the term is used generally.

It depends on what you define as a local maximum here. If local maxima can only exist within the interior of the interval, then yes, in general the maximum of the local maximum is not the global maximum. However, if a local maximum is defined to occur at a point $x^*$ where $f(x^*) \geq f(x)$ for all $x$ in a neighborhood of $x^*$, then it follows that the global maximum is a local maximum, and must be the greatest among these.

$\endgroup$
1
  • $\begingroup$ Thank you for your answer. Could you please explain your last sentence again? I don't quite understand how the 2nd definition follows from that... or is "neighborhood" only valid in the interior parts of the interval? $\endgroup$
    – FGSUZ
    Feb 7, 2021 at 10:49

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .