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01Increase, Decrease, and Concavity

01Increase, Decrease, and Concavity - INCREASE DECREASE AND...

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I NCREASE , D ECREASE , AND C ONCAVITY Definition : Let f be defined on an interval, and let 1 x and 2 x denote numbers in that interval. a) f is increasing on the b) f is decreasing on the c) f is constant on the interval if ) ( ) ( 2 1 x f x f < interval if ) ( ) ( 2 1 x f x f interval if ) ( ) ( 2 1 x f x f = whenever 2 1 x x < whenever 2 1 x x < for all 1 x and 2 x ) ( 1 x f ) ( 2 x f ) ( 2 x f ) ( 1 x f ) ( 2 x f ) ( 1 x f 1 x 2 x 1 x 2 x 1 x 2 x Theorem : Let f be a function that is continuous on a closed interval [ a, b ] and differentiable on the open interval ( a, b ). a) If 0 ) ( ' x f for b) If 0 ) ( ' < x f for c) If 0 ) ( ' = x f for every value of x every value of x every value of x in ) , ( b a then f is in ) , ( b a then f is in ) , ( b a then f is increasing on [ a, b ]. decreasing on [ a, b ]. constant on [ a, b ]. positive slope negative slope zero slope Ex . Find the intervals for which the following functions are increasing and/or decreasing. 1) 3 4 ) ( 2 + - = x x x f ) 2 ( 2 4 2 ) ( ' - = - = x x x f 2) 3 ) ( x x f = 2 3 ) ( ' x x f =

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3) 2 12 4 3 ) ( 2 3 4 + - + = x x x x f Interval ) 2 )( 1 )( ( 12 + - x x x ) ( ' x f Conclusion
Direction of Curvature: Concave down Concave up Definition

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