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# Chapter12.2 - 1 9 6 2 3-= x x x x f 9 12 3 1 9 6 2 2 3-=-=...

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Chapter 12.2 Page 664 Definition: (i) If ) ( ' x f is increasing over ( a , b ), then f is concave up on ( a , b ). (ii) If ) ( ' x f is decreasing over ( a , b ), then f is concave down on ( a , b ). (See Figure 2) Page 665 Notation: The second derivative of f , written ( 29 ' ) ( ' ) ( ' ' x f x f = , or y’’ = ( y’ ) or = dx dy dx d dx y d 2 2 Page 666 Example 1. (C) 3 ) ( x x h = 2 3 3 )' ( ) ( ' x x x h = = x x x h 6 )' 3 ( ) ( ' ' 2 = = Page 667 Definition: An inflection point is a point where the graph changes from upward to downward or from downward to upward. 1

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Remark: If f is continuous at c and c is an inflection point, then 0 ) ( ' ' = c f . Page 668 Example 2: Find the inflection point(s) of
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Unformatted text preview: 1 9 6 ) ( 2 3 + +-= x x x x f . 9 12 3 )' 1 ( )' 9 ( )' 6 ( )' ( ) ( ' 2 2 3 +-= + +-= x x x x x x f 12 6 )' 9 ( )' 12 ( )' 3 ( ) ( ' ' 2-= +-= x x x x f Solve ⇐ = =-= 2 12 6 ) ( ' ' x x x f f f point of inflection. Page 671 Example 5: Find the inflection point(s) of 3 4 2 ) ( x x x f-= . x x x f x x x f 12 12 ) ( ' ' 6 4 ) ( ' 2 2 3-=-= Solve ⇐ = = =-=-1 , ) 1 ( 12 12 12 2 x x x x x x f f points of inflection 2...
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Chapter12.2 - 1 9 6 2 3-= x x x x f 9 12 3 1 9 6 2 2 3-=-=...

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