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The definition(wiki): A real-valued function $f$ defined on a domain $X$ has a global (or absolute) maximum point at $x∗$, if $f(x∗) \geq f(x)$ for all $x$ in $X$.

My question is can there be more than one global maximum(or minimum) of a function?

The definition does not suggest that there is only 1 global maximum(or minimum). Example: let $f(x) = x^2$ on the domain $X = [-2,2]$, thus the global maximum is $f(-2) = 4 = f(2)$ as by definition $f(-2) = f(2) \geq f(x) \ \forall x\in X$, thus we have 2 global maiximums.

I tried to confirm this but there are so many sorces saying that there can only be one: Example1, Example2.

Am I misunderstanding something? I would appreciate any insight you can offer.

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    $\begingroup$ I'm editing your tag because extremal graph theory is something different (though it sounds enticing). $\endgroup$
    – Randall
    Commented May 15 at 14:21
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    $\begingroup$ I'd say Example 1: never says there can only be one global maximum point but only in the examples considered cases where there were one, incorrectly implies it. Example 2: says there can only be one maximum global value (Example one [which you quoted] refers to maximum points) There can only be one maximum value for obvious reasons. But whereas Example 1 was merely short-sighted Example 2 is being negligent. In stressing there is only one it never says why and all examples imply it is refering to points. $\endgroup$
    – fleablood
    Commented May 15 at 15:44
  • $\begingroup$ The more I read Example 2: the more I think it is just plain wrong. It points out that although the local maximum (points) can appear in many places it (value) needs only be largest of the (values of) points in a limited range. A global maximum must have the largest value of all locations. Then example 2 wrongly assumes such a global maximum can only occur in one location. $\endgroup$
    – fleablood
    Commented May 15 at 15:55

2 Answers 2

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The term "maximum" applies to the $y$-coordinate or output of the function. In your case, $4$ is the sole largest $y$-coordinate, though it appears twice. So the location of the maximum $y$ need not be unique, but the $y$-value will be unique.

Note that in your definition, you have a "maximum point at $x^*$." This does not say that $x^*$ is the maximum. (And it's not, that would be $f(x^*)$.)

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    $\begingroup$ Thank you for your answer, I believe I misunderstood what they meant only 1 maximum. I was thinking of points of where maximum occurs, but they were talking of like you say on the y-axis. $\endgroup$
    – Reuben
    Commented May 15 at 14:22
  • $\begingroup$ Common initial misunderstanding. $\endgroup$
    – Randall
    Commented May 15 at 14:22
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    $\begingroup$ Does it? The OP used the phrase "maximum point". So the question arises: Is the phrase "maximum point" a valid and legitimate and if so what does it mean? I'd say "maximum" or "maximum value" meaning the value (which must be singular as any to values must be unequal and thus one larger than the other) and "maximum location" meaning elements of the domain, and "maximum point" meaning an ordered pair, is both common and legitimate. $\endgroup$
    – fleablood
    Commented May 15 at 15:30
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    $\begingroup$ I’m very familiar with mathematicians from several fields using “maximum” to mean either the point or just the function value, depending on context — saying both things like “the function has several maxima” and “this maximum is attained at several different points”. So I wouldn’t call OP’s usage a misunderstanding at all. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented May 15 at 23:06
  • $\begingroup$ But it is a common misunderstanding to think that a function can have more than one maximum value by misinterpreting "value" to mean "location." I've taught calculus for 30 years and this comes up nearly every time. Note that this is precisely OP's source of confusion, as OP's sources clearly state that a function has a unique global maximum, if it has one at all. In fact, he says this in the first comment above. $\endgroup$
    – Randall
    Commented May 16 at 0:02
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Actually, "there's only one maximum" doesn't necessarily mean there's only one maximum point.

You can have infinitely many maximum points - consider $f(x)=\cos x$.

You can also have uncountably many maximum points - consider $f(x)=-|x|-|x-1|$.

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  • $\begingroup$ ... or just $f(x)=0$, which has a maximum and minimum at every point. $\endgroup$
    – Hagen von Eitzen
    Commented May 16 at 12:22

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