5odd - 5 GRAPHING AND OPTIMIZATION EXERCISE 5-1 Things to...

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EXERCISE 5-1 215 5 GRAPHING AND OPTIMIZATION EXERCISE 5-1 Things to remember: 1 . INCREASING AND DECREASING FUNCTIONS For the interval ( a , b ): f '( x ) f ( x ) Graph of f Examples + Increases Rises - Decreases Falls 2 . CRITICAL VALUES The values of x in the domain of f where '( ) = 0 or where '( ) does not exist are called the CRITICAL VALUES of . The critical values of are always in the domain of and are also partition numbers for ', but ' may have partition numbers that are not critical values. If is a polynomial, then both the partition numbers for ' and the critical values of are the solutions of '( ) = 0. 3 . LOCAL EXTREMA Given a function . The value ( c ) is a LOCAL MAXIMUM of if there is an interval ( m , n ) containing such that ( ) ! ( ) for all in ( , ). The value ( e ) is a LOCAL MINIMUM of if there is an interval ( p , q ) containing such that ( ) " ( ) for all in ( , ). Local maxima and local minima are called LOCAL EXTREMA. A point on the graph where a local extremum occurs is also called a TURNING POINT. 4 . EXISTENCE OF LOCAL EXTREMA If is continuous on the interval ( a, b ), c is a number in ( ) and ( ) is a local extremum, then either ’( ) = 0 or ’( ) does not exist (is not defined).
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216 CHAPTER 5 GRAPHING AND OPTIMIZATION 5 . FIRST DERIVATIVE TEST FOR LOCAL EXTREMA Let c be a critical value of f [ ( ) is defined and either '( ) = 0 or '( ) is not defined.] Construct a sign chart for '( x ) close to and on either side of . f ( c ) is a local minimum. If f' ( x ) changes from negative to positive at c , then f ( c ) is a local minimum. f ( c ) is a local maximum. If f' ( x ) changes from positive to negative at c , then f ( c ) is a local maximum. f ( c ) is not a local extremum. If f' ( x ) does not change sign at c , then f ( c ) is neither a local maximum nor a local minimum. f ( c ) is not a local extremum. If f' ( x ) does not change sign at c , then f ( c ) is neither a local maximum nor a local minimum. Sign Chart f ( c ) - - - + + + f' ( x ) f ( x ) Decreasing Increasing x m c n ) ( + + + - - - f' ( x ) f ( x ) Increasing Decreasing x m c n ) ( - - - - - - f' ( x ) f ( x ) Decreasing Decreasing x m c n ) ( + + + + + + f' ( x ) f ( x ) Increasing Increasing x m c n ) ( 6 . INTERCEPTS AND LOCAL EXTREMA FOR POLYNOMIAL FUNCTIONS If ( ) = a n + -1 + … + 1 + 0 , # 0 is an th degree polynomial then has at most intercepts and at most -1 local extrema. 1. ( a, b ), ( d, f ), ( g, h ) 3. ( b, c ), ( c, d ), ( f, g ) 5. = c, d, f 7. = b, f 9. has a local maximum at = , and a local minimum at = ; does not have a local extremum at = b or at = d . 11. e 13. 15. 17. 19. ( ) = 2 2 - 4 ; domain of : (- $ , $ ) '( ) = 4 - 4; ' is continuous for all . '( ) = 4 - 4 = 0 = 1 Thus, = 1 is a partition number for ', and since 1 is in the domain of , = 1 is a critical value of .
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EXERCISE 5-1 217 Sign chart for f ': Increasing Decreasing - - - - + + + + '( x ) ( ) Local minimum 0 1 2 Test Numbers " f ( ) 0 # 4( # ) 2 + ) Therefore, f is decreasing on (- $ , 1); is increasing on (1, $ ); (1) = -2 is a local minimum. 21. ( ) = -2 2 - 16 - 25; domain of : (- $ , $ ) '( ) = -4 - 16; ' is continuous for all and '( ) = -4 - 16 = 0 = -4 Thus, = -4 is a partition number for ', and since -4 is in the domain of , = -4 is a critical value for . Sign chart for ': '( ) ( ) + + + + - - - - - -5 -4 0 Test Numbers '( ) ! 5 + ) 0 ! 16( ! ) Therefore, is increasing on (- $ , -4); is decreasing on (-4, $ ); has a local maximum at = -4.
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This note was uploaded on 04/21/2009 for the course MATH 121 taught by Professor Hamidy during the Spring '09 term at Miramar College.

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5odd - 5 GRAPHING AND OPTIMIZATION EXERCISE 5-1 Things to...

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