ࡱ> =?< (abjbj *drrQ <<<*($ !#9":,M0yR q0*RZ$atZ$MZ$M$*Z$< \: What f ' and f " Tell us About f Assume f is defined everywhere. Behavior of f ' ( (implies)Behavior of ff ' > 0 on an intervalf is increasing on the intervalf ' < 0 on an intervalf is decreasing on the intervalf ' (c) = 0f has a horizontal tangent at x = cf ' (x) < 0 for x < c and f ' (x) > 0 for x > c That is, f ' changes from negative to positive at c f has a relative minimum at x = cf ' (x) > 0 for x < c and f ' (x) < 0 for x > c That is, f ' changes from positive to negative at c f has a relative maximum at x = cf ' increasing (or f " > 0) on an intervalf is concave up on the intervalf ' decreasing (or f " < 0) on an intervalf is concave down on the intervalf " changes sign at c ALSO f' goes from increasing to decreasing, or vice versa, at cf has an inflection point at x = c In proving a relative minimum or maximum, it is never enough to show that the derivative is zero. A single example demonstrates this: Consider C(x) = x3 for x = 0. The derivative C'(x) = 3x2 so C'(0) = 0 but the function has neither a minimum or maximum there. To prove a relative maximum or minimum when x = c, it is always necessary to do one of these things: Show that f ' (x) changes sign at x = c (i.e use the First Derivative Test) Or Show that f '(c)=0 and f "(c) `" 0 (i.e. use the Second Derivative Test) Values of x where f ' (x) = 0 or f '(x) is undefined are called critical numbers. These are merely candidates for x-values of a maximum or minimum  you must still see if f ' (x) changes sign. Even well-labeled sign charts are not enough to show extrema and inflection points. You must state the link between f ' and f. See "Lessons Learned at the 2005 Readings." If the problem asks for an absolute extreme, you must also evaluate the function at the endpoints of the interval, and compare. Old Exam Questions on f, f ' and f( YearNumberFunction PresentationFunction TypePart2005AB4(b)2003AB3graphical2000AB3graphical1997AB4(b)1982AB2(e)1981AB3 BC1(e)1980AB5 BC2(d)1980BC7(e)1979AB6(a) (c)1978AB2(d)1978BC2(c)1977AB2(d)1977BC1(a)1976AB5 BC3(b)1975AB4 BC1(c)1974AB1 BC1analytictrigonometric(a)1971AB7 BC3(c)1970AB2graphicalgeneral(a) (b) (c) (d) (e)1970AB3 BC2(e)1969AB3 BC3(c)1969AB7analyticexponential(a)1969BC1analytictrigonometric(a) Using the Tangents Program (TI83/84 only) The program Tangents is a great way to see the relationship between f and f '. 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