A quantityIf you writeucan only take a maximum or minimum value when its differential duis zero or undefined.

However, *u* might not have a maximum or minimum value!
Assuming that *u* varies continuously
(which it must if Calclulus is to be useful at all),
then it must have a maximum and minimum value
whenever the range of possibilities is *compact*;
this means that if you pass continuously through the possibilities in any way,
then you are always approaching some limiting possibility.
However, if the range of possibilities heads off to infinity in some way,
or if there is an edge case that's not quite possible to reach,
then you also have to take a limit to see what value *u* is approaching.
If any such limit is larger than every value that *u* actually reaches
(which includes the possibility that a limit is ∞),
then *u* has no maximum value;
if any such limit is smaller than every value that *u* actually reaches
(which includes the possibility that a limit is −∞),
then *u* has no minimum value.

So in the end, you look at these possibilities:

- when the derivative of
*u*is zero or undefined, - the extreme edge cases, and
- the limits approaching impossible limiting cases.

Go back to the course homepage.

This web page was written in 2015 by Toby Bartels, last edited on 2015 August 14. Toby reserves no legal rights to it.

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