f c If f x 0 both on some interval a c and f x 0 on some interval c b then x c

# F c if f x 0 both on some interval a c and f x 0 on

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f . (c) If f ’( x ) 0 both on some interval ( a , c ) and f ’( x ) 0 on some interval ( c , b ), then x = c is not a local extreme point for f . 8.6 Example Classify the stationary points of So x =-1 and x =2 are the stationary points. 3 2 1 1 2 ( ) 1, ( ) ( , ) 9 6 3 f x x x x D f = - - + = -∞ +∞ = -∞ +∞ 2 1 1 2 '( ) 3 3 3 f x x x = - - 2 1 ( 2) 3 x x = - - - 1 ( 1)( 2) 3 x x = + -

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3 8.6 Example The signdiagram for f ’( x ) is: Conclusion: x =-1 is a local maximum point x =2 is a local minimum point NO GLOBAL max or min! 1 '( ) ( 1)( 2) 3 f x x x = + - 8.6 Theorem: second-derivative test for local extreme points Let f be a twice differentiable function in an interval I and let c be an interior point of I . Then: (a) If f ’( c ) = 0 and f ’’( c ) < 0 then x is a local maximum point . (b) If f ’( c ) = 0 and f ’’( c ) > 0 then x is a local minimum point . (c) If f ’( c ) = 0 and f ’’( c ) = 0 ? (Try to use the first-derivative test) 8.6 Theorem: second-derivative test for local extreme points If f ’( c ) = 0 and f ’’( c ) = 0 ? f ’(0) = f ’’(0) = 0 and 0 is a minimum point 8.6 Theorem: second-derivative test for local extreme points If f ’( c ) = 0 and f ’’( c ) = 0 ? f ’(0) = f ’’(0) = 0 and 0 is a maximum point 8.6 Theorem: second-derivative test for local extreme points If f ’( c ) = 0 and f ’’( c ) = 0 ? f ’(0) = f ’’(0) = 0 and 0 is an inflection point 8.7 Inflection points Points at which a function f changes from convex ( f ’’( x ) 0) to concave ( f ’’( x ) 0), (or vice versa), are called inflection points .
4 8.7 Example Find the possible inflection points for 6 4 ( ) 10 f x x x = - - 5 3 '( ) 6 40 f x x x = - = - = = 4 2 ''( ) 30 120 f x x x = - 2 2 30 ( 4) x x = - 2 30 ( 2)( 2) x x x = - + 0 for 0, 2, 2 x x x = = = = - = = = = - = = 8.7 Example Signdiagram is as follows f ’’ changes sign at x =-2 and at x =2 so these are the inflection points.

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