f x x c f x x c x c f 83 Economic example Theorem 1a A profit function is given

F x x c f x x c x c f 83 economic example theorem 1a

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8.3 Economic example Theorem 1a A profit function is given by: Find the amount N which maximizes profit in this case. So N = 100 maximizes profit. 1 2 ( ) 10 0.5 , N 0 N N N π = - = - 1 2 5 '( ) 5 0.5 0.5 0 N N N π - = - = - = 100 N = '( ) 0 when 100 N N π > < '( ) 0 for 100 N N π < > < 8.3 Same example: now using theorem 2 A profit function is given by: Find the amount N which maximizes profit in this case. is concave So N = 100 maximizes profit. 1 2 ( ) 10 0.5 , N 0 N N N π = - = - 1 2 '( ) 5 0.5 N N π - = - 1 1 2 1 2.5 ''( ) 2 0 2 N N N N π - - = - = < = < π 8.4 The extreme-value theorem THEOREM : If the function f is a continuous function over a closed bounded interval [ a , b ] then there exists a point d in [ a , b ] where f has a minimum and a point c in [ a , b ] where f has a maximum. ( ) ( ) ( ) for all in [ , ]. f d f x f c x a b so the extreme-value theorem assures us that maximum and minimum points do exist. 8.4 Candidate extreme points Extreme points of an arbitrary function can be one of the following types: (a) Interior point in I where f ’( x ) = 0 (b) End points of I (if included in I ) (c) Interior point in I where f ’ does not exist Points satisfying any of these three conditions will be candidate extreme points . 8.4 Problem Problem: Find the maximum and minimum values of a differentiable function f defined on a closed bounded interval [ a , b ]. (i) Find all the stationary points of f in ( a , b ) – that is find all points x in ( a , b ) that satisfy the equation f ’( x ) = 0.
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