Problem: Find the global extrema of

on the interval I = [1,∞).

Solution: To find global extrema over an interval we compare values at suspicious points; however, at suspicious points that are open endpoints of the given set I we use limit to find the relevant value. One kind of suspicious points is the endpoints, here 1 and infinity (where we use limit). The other kind is critical points of the given function. To find these we first differentiate the given function using the usual rules.

Critical points are points of the domain where the derivative does not exists (there are none here) and points where the derivative is zero, which here means x = −2 and x = 2. However, the former does not belong to the set I and therefore it is irrelevant.

We have three candidates for global extrema, namely 1, 2, and infinity. We compare corresponding values:

The largest value is 1/4 and it is attained, therefore we have a global maximum there, max _I ( f ) = f (2) = 1/4. The smallest value is 0, but it is reached only by a limit, not at some proper point of I, which means that there is no global minimum of f on I. By the way, it shows that inf _I ( f ) = 0.

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