# 43web - Definition A function 0 is increasing on an...

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MTH 173 setion 4.3 notes, page 1 Definition A function is on an interval if for any two numbers and 0 B B increasing " # in the interval, implies . B  B 0 B  0 B " # " # a b a b A function is on an interval if for any two numbers and 0 B B decreasing " # in the interval, implies . B  B 0 B  0 B " # " # a b a b Thm Let be a function that is continuous on the closed interval and 0 +ß , c d differentiable on the open interval . a b +ß , 0 B  ! B +ß , 0 +ß , If for all in , then is increasing on . w a b a b c d 2 If for all in , then is decreasing on . Þ 0 B  ! B +ß , 0 +ß , w a b a b c d 3 If for all in , then is constant on . Þ 0 B o ! B +ß , 0 +ß , w a b a b c d Proof: Let be in the interval such that . B ß B +ß , + Ÿ B  B Ÿ , " # " # a b By the MVT, there is a where such that - B  -  B " # 0 - o w 0 B 0 B B B a b a b a b # " # " . By hypothesis, 0 -  !Þ w a b Since we know . B  B ß B  B  ! " # # " Thus in , the LHS and the denominator on the RHS 0 - o 0 B  0 B B  B w # " # " a b a b a b are positive. Then the numerator must also be positive. 0 B  0 B a b a b # " Thus 0 B  0 B  ! Ê 0 B  0 B Þ a b a b a b a b # " " # Since and are any two numbers in B B " # a b +ß , , we know the function increases everywhere in the interval.

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MTH 173 setion 4.3 notes, page 2 Thm Let be a critical number of a function on an open interval containing . If - 0 M - 0 is differentiable on the interval, except possibly at , then can be classified - 0 - a b as follows: 1. If changes from negative to positive at then is a
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43web - Definition A function 0 is increasing on an...

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