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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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    (In Italian this concept is called "punto estremante", the English Wikipedia article suggests no equivalent phrasing.)2011-12-23
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    Supremum and maximum are not quite the same thing - the supremum can exist when a maximum does not.2011-12-23
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    @ThomasAndrews I am very well aware. Did I make that confusion anywhere in the page?2011-12-23

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Often one writes $\displaystyle \operatorname*{argmax}_x f(x)$ for the value of $x$ that maximizes $f(x)$.

(Despite the fact that I have frequently seen that notation for many years, I find that it's not a standard TeX operator name; one cannot type \argmax and have it understood by TeX.)

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    `\arg\max` seems to work.2011-12-23
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    @badp I believe [your edit suggestion](http://math.stackexchange.com/suggested-edits/3641) was made with good intentions, but I rejected it because it changed the answer too heavily. I suggest you write it as a comment instead, so that Michael can introduce that change if necessary.2011-12-23
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    @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 $\operatorname{argmax}$.2011-12-23
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    @Srivatsan May I point you to the comment I _have_ added? His post could've been improved with a comment in Michael's inbox and no further action on his part, now he's going to get this frankly uninteresting discussion in his inbox instead. Nevermind edit suggestions typically are rejected for being _too minor_...2011-12-23
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    http://en.wikipedia.org/wiki/Argmax2011-12-23
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    @MichaelHardy ...meh, I didn't think that would be a problem.2011-12-23
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    [Wikipedia](https://en.wikipedia.org/wiki/Arg_max) uses: `\underset {x}{\operatorname {arg\,max} }` giving: $\underset {x}{\operatorname {arg\,max} }$2017-08-04
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    \displaystyle \operatorname*{argmax}_x yields $\displaystyle \operatorname*{argmax}_x.$ That doesn't work in Wikipedia but it works here and it works in LaTeX. And of course in a "displayed" context one need not use \displaystyle: $$ \operatorname*{argmax}_x $$ I have now edited this answer accordingly. $\qquad$2017-08-04
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    @TomHale : $\uparrow$ (Just in case you missed this.)2017-08-10