# L24 - 10 Find all inﬂection points of the graph of f ( x...

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L24 – Derivatives and the Shape of the Curve A function f is increasing (decreasing) on an interval if on that interval 1

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Find the intervals on which f ( x ) = x 4 - 4 3 x 3 is increasing and decreasing. 2
Where is g ( x ) = x 2 - 1 x 4 increasing and decreasing? 3

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First Derivative Test – Suppose that c is a critical number of a continuous function f . If f 0 changes from positive to negative at c , then If f 0 changes from negative to positive at c , then If f 0 does not change signs at c , then 4
Find all local extrema of f ( x ) = x 2 3 ( x - 5). 5

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Concavity Def – If the graph of f lies ( ) its tangents on an interval I , then it is concave ( ) on I . 6
Graphically Where is f ( x ) concave up and concave down? 7

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Test for Concavity If ( ) for all x on interval I , then the graph of f is concave ( ) on I . Why? Consider slope of tangent lines. 8
Def – A point P on a curve is called an if concavity changes from up to down or down to up at P . Identify the inﬂection points on the graph above. 9

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Determine when the graph of f ( x ) = x 4 - 4 3 x 3 is concave up and concave down, and ﬁnd any inﬂection points. Make a rough sketch.

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Unformatted text preview: 10 Find all inﬂection points of the graph of f ( x ) = e-x 2 . Make a rough sketch. 11 Second Derivative Test – Suppose f 00 ( x ) is continuous near c . a) If f ( c ) = 0 and f 00 ( c ) then f has a local at c . b) If f ( c ) = 0 and f 00 ( c ) then f has a local at c . Determine the local extrema of the functions above. 12 Find all relative extrema and inﬂection points of the graph of f ( x ) = 3 x 1 3-x and make a rough sketch of f ( x ). 13 Find all relative extrema and inﬂection points of the graph of f ( x ) = 2 5 x 5 3-x 2 3 . Make a rough sketch of f ( x ). 14 Given the graph of f ( x ), determine the following: a) Intervals on which f is increasing/decreasing b) Local extrema c) Intervals on which f is concave up/down d) Points of inﬂection e) If f (0) = 3, make a possible sketch of f ( x ). 15...
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## This note was uploaded on 01/13/2010 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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L24 - 10 Find all inﬂection points of the graph of f ( x...

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