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Are there any rules concerning when to use the word 'extreme' and when to use 'extremal'? For instance, in a convex set you speak of an 'extreme point', while there is also usage of 'convex extremal functions'.

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In my opinion one use the word 'extremal' in the sense of calculus of variation. So if you have to minimize/maximize a functional over some functions or curves. But if you maximize or minimize over a set of points you speak about 'extreme'.

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