Finding Absolute Extrema

    Absolute extrema are defined and a procedure for finding absolute extrema on a given closed bounded interval is given. We give a couple of examples to illustrate this procedure.

Proposition (Procedure for Finding Absolute Extrema) Let be a continuous function whose domain contains Then to find the absolute extrema of do the following:

    (i) Find all critical numbers of on and evaluate for each of these numbers.

    (ii) Evaluate at the boundary; that is, find and

    (iii) Select the largest and smallest values those listed in (i) and (ii). The largest is the absolute maximum and the smallest is the absolute minimum.

Example (Finding Absolute Extrema) Find the absolute extrema of on the given interval for

on the interval
    
    Solution. We have, and so the critical numbers are Therefore, we compute
    


Here is a graph of on the interval

Example (Finding Absolute Extrema) Find the absolute extrema of on the given interval for

on the interval
    
    Solution. We have, and so the critical number is Therefore, we compute
    

Example (Finding Absolute Extrema) Find the absolute extrema of on the given interval for

on the interval

    Solution. We have, and so there are no critical numbers because is not in our domain of . Therefore, we compute
    

Example (Finding Absolute Extrema) Find the absolute extrema of on the given interval for

on the interval

    Solution. We have, and so to find our critical number we need to solve or which is of course where is any integer. However, for our given domain of   the only critical number is
    


Here is a graph of on the interval

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