# What's a correct symbolism for “value that maximizes” [duplicate]

Possible Duplicate:
Math notation for location of the maximum

Given a function $f(x)$, we can normally find $\max_i f(i)$. This expression evaluates to the maximum value of $f(x)$. Sometimes, however, what is interesting isn't as much the maximum value itself, but the value at which the function reaches its maximum, whatever that might be:

$$i:f(i)=\max_t f(t)$$

In plain English, I want to know who has eaten the most $f$ruit, rather than how much he's eaten.

This is kind of cumbersome, and perhaps needlessly requires a new symbol t. Besides, this isn't entirely correct, as $f(x)$ could very well have more than one maximum:

$$I=\{i:f(i)=\max_tf(t)\}$$

Is there a nicer way to express this concept?

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## marked as duplicate by t.b., Srivatsan, J. M., Ilya, Asaf KaragilaDec 23 '11 at 13:25

(In Italian this concept is called "punto estremante", the English Wikipedia article suggests no equivalent phrasing.) – badp Dec 23 '11 at 0:15
Supremum and maximum are not quite the same thing - the supremum can exist when a maximum does not. – Thomas Andrews Dec 23 '11 at 0:41
@ThomasAndrews I am very well aware. Did I make that confusion anywhere in the page? – badp Dec 23 '11 at 0:43

Often one writes $\underset{x}{\operatorname{argmax}} f(x)$ for the value of $x$ that maximizes $f(x)$.
\arg\max seems to work. – badp Dec 23 '11 at 0:18
@badp : I see that you suggested \arg\max. Let's try that: $\displaystyle\arg\max_x f(x)$. The subscript $x$ ends up directly below $\max$ rather than symmetrically under $\operatorname{argmax}$. – Michael Hardy Dec 23 '11 at 0:31