Relative Extrema

    Now we define relative extrema and state the relative extrema theorem. If a function is defined on an open interval and if at some point in that interval the function reaches a maximum or minimum value (relative to that interval), then we say that the function has a relative extrema on that interval. A maximum or minimum value that occurs at an endpoint is not, by definition, a relative maximum nor a relative minimum. A relative maximum or relative minimum must occur in the interior of an interval.

Definition (Relative Maximum) Let be a function defined on an open interval If and for all in then is called a relative maximum of on A relative maximum is sometimes called a local maximum.

Definition (Relative Minimum) Let be a function defined on an open interval If and for all in then is called a relative minimum of on A relative minimum is sometimes called a local minimum.

Definition (Relative Extrema) Let be a function defined on an open interval If and   is either a relative maximum or a relative minimum then is a relative extrema, and we say that is a relative extreme value.

    The following proposition is often called Fermat's Theorem due to acknowledgment that Fermat realized the result first. The following examples show that even when there need not be a maximum or minimum at In other words, the converse of Fermat's Theorem is false in general. Furthermore, there may be an extreme value when or when does not exist.

Proposition (Relative Extrema Theorem) If has a relative extremum at and exists then

    Proof. Since is differentiable at must be positive, zero, or negative. Suppose



Then there exists an interval containing such that for all in This produces the following inequalities for - values in the interval If and then is not a relative minimum. If and then is not a relative maximum. So the assumption that leads to a contradiction. Assuming that will also lead to a similar contradiction. Thus it must be the case as desired.

Example (Relative Extrema) Determine if the relative extrema theorem applies and if so find the relative extrema for the function:



    Solution. The function has its minimum value (local and absolute) at but we can not find this absolute minimum by setting because is not defined at      

Determine if the relative extrema theorem applies and if so find the relative extrema for the function:  


    
    Solution. Since we have However, since does not have a relative extremum at the converse of the relative extrema theorem does not hold.

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Cite this as:
Relative Extrema
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/relative-extrema.html