What f ' and f' Tell us About f
What f ' and f " Tell us About f
Assume f is defined everywhere.
|Behavior of f ' ( (implies) |Behavior of f |
|f ' > 0 on an interval |f is increasing on the interval |
|f ' < 0 on an interval |f is decreasing on the interval |
|f ' (c) = 0 |f has a horizontal tangent at x = c |
|f ' (x) < 0 for x < c and | |
|f ' (x) > 0 for x > c |f has a relative minimum at x = c |
|That is, f ' changes from negative to positive at c | |
|f ' (x) > 0 for x < c and | |
|f ' (x) < 0 for x > c |f has a relative maximum at x = c |
|That is, f ' changes from positive to negative at c | |
|f ' increasing (or f " > 0) on an interval |f is concave up on the interval |
|f ' decreasing (or f " < 0) on an interval |f is concave down on the interval |
|f " changes sign at c ALSO |f has an inflection point at x = c |
|f' goes from increasing to decreasing, or vice versa, at 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(
|Year |Number |Function Presentation|Function Type |Part |
|2005 |AB4 | | |(b) |
|2003 |AB3 |graphical | | |
|2000 |AB3 |graphical | | |
|1997 |AB4 | | |(b) |
|1982 |AB2 | | |(e) |
|1981 |AB3 BC1 | | |(e) |
|1980 |AB5 BC2 | | |(d) |
|1980 |BC7 | | |(e) |
|1979 |AB6 | | |(a) (c) |
|1978 |AB2 | | |(d) |
|1978 |BC2 | | |(c) |
|1977 |AB2 | | |(d) |
|1977 |BC1 | | |(a) |
|1976 |AB5 BC3 | | |(b) |
|1975 |AB4 BC1 | | |(c) |
|1974 |AB1 BC1 |analytic |trigonometric |(a) |
|1971 |AB7 BC3 | | |(c) |
|1970 |AB2 |graphical |general |(a) (b) (c) (d) (e) |
|1970 |AB3 BC2 | | |(e) |
|1969 |AB3 BC3 | | |(c) |
|1969 |AB7 |analytic |exponential |(a) |
|1969 |BC1 |analytic |trigonometric |(a) |
Using the Tangents Program (TI83/84 only)
The program Tangents is a great way to see the relationship between f and f '. Some suggestions:
• Run it first with the #1 function on the menu, a carefully chosen cubic.
• You can Pause the program by pressing "+". Resume with .
• Run it with the #3 function, sin(x), to show [pic][pic]
• Run it with the #4 function, sin(2x), to illustrate the chain rule.
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