The key principle of optimization is this:
A quantity u can only take a maximum or minimum value when its differential du is zero or undefined.
If you write u as f(x), where f is a fixed differentiable function and x is a quantity whose range of possible values you already understand, then du = f'(x) dx. So u can only take an extreme value when its derivative (with respect to x) is zero or undefined or when you can no longer vary x smoothly however you please (which must occur at the extreme values of x and typically only then).

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:

Whichever of these has the largest value of u gives you the maximum, and whichever has the smallest value of u gives you the minimum; but if the largest or smallest value is only approached in the limit, then the maximum or minimum technically does not exist.
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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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