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Unformatted text preview: 17. The First Derivative Test for Local Extrema P. K. Lamm 10/24/11 (22:00) p. 1 / 7 Lecture Notes: 17. The First Derivative Test for Local Extrema These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Local Maximum and Minimum Values of Functions In an earlier set of notes we looked at the concept of absolute extrema (absolute maximum and absolute minimum values) of a function f . But there are other points which are still of interest to us and, in particular, give information about how to graph a function. For example, in the graph below, the function on [- 2 , 4] clearly has an absolute max at the endpoint x =- 2 and an absolute min at the endpoint x = 4. But at x = 0 and x = 2 there is additional behavior that is worth marking. These are points of local or relative extrema of the function f . Definition: Suppose f is defined on a given domain D (an open, closed, or half-closed interval in the real line), and that c is in the interior of D . We say f ( c ) is a local or relative maximum of f on D if there exists an open interval I containing c such that f ( c ) ≥ f ( x ) , for all x ∈ I ∩ D. (Note that we do not require f ( c ) ≥ f ( x ) for all x ∈ D .) We say f ( c ) is a local or relative minimum of f on D if there exists an open interval I containing c such that f ( c ) ≤ f ( x ) , for all x ∈ I ∩ D. We say f ( c ) is a local or relative extremum if it is either a local maximum or a local minimum. Note: This definition can be used for local extrema at points c in the interior of the interval D , and also for the endpoints a , b of a closed interval D = [ a,b ]. For example, if D = [ a,b ], then f has a local max at the point x = a if there is some d ∈ ( a,b ] for which f ( a ) ≥ f ( x ), for all x ∈ ( a,d ). Theorem: If f be defined on a given domain D . If f has a local extremum at an interior point c ∈ D , then either f ( c ) = 0 or f ( c ) is undefined. 1 17. The First Derivative Test for Local Extrema P. K. Lamm 10/24/11 (22:00) p. 2 / 7 That is, if f ( c ) is a relative extremum of f for c an interior point of [ a,b ], then c is a critical point of f . Note: The converse is not necessarily true: that is, critical points need not be points where f has a relative extremum. We will see this situation in Examples 2.1–2.4 below....
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