x 0 is not an inflection point since f does not change sign at x 0 2 30 2 2 f x

# X 0 is not an inflection point since f does not

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so these are the inflection points. x =0 is not an inflection point since f ’’ does not change sign at x =0. 2 ''( ) 30 ( 2)( 2) f x x x x = - + Continuity lim ( ) ( ) x a f x f a = Definition: The function f(x) is continuous in x=a if The following conditions must all be fulfilled in order for f to be continuous at x= a : (i) The function f must be defined at x = a . (ii) The limit must exist. (iii) L= f ( a ). lim ( ) x a f x L = Def: The function f(x) is discontinuous in x=a if f(x) is not continuous in x=a. Def: The function f(x) is continuous over the interval [a,b] if f(x) is continuous in every point in [a,b]. 8.5 Economic example II Assume that the price per unit P (Q) varies with Q according to the formula: Suppose the cost function is: The profit is: 3 2 1 1 1000 ( ) 50 600 3 3 C Q Q Q Q = - + + - ( ) ( ) ( ) ( ) Q R Q C Q PQ C Q π = - = - = - 1 ( ) 100 , [0,300] 3 P Q Q Q = - 3 2 1 1 1 1000 (100 ) ( 50 ) 3 600 3 3 Q Q Q Q Q = - - - + + - 3 1 1000 50 600 3 Q Q = - + - + 8.5 Economic example II Find the production level that maximizes profit and compute the maximum profit. The candidate extreme points are Q =100 and the endpoints Q =0, Q =300. So Q =100 maximizes profit and that maximum profit is 3000. 100 Q = 1000 91000 (0) , (100) 3000, (300) 3 3 π π π π π π π π - - = = = = = = 3 1 1000 ( ) 50 600 3 Q Q Q π = - + - = - + - 2 1 '( ) 50 0 200 Q Q π = - + = = Continuity lim ( ) ( ) x a f x f a = Definition: The function f(x) is continuous in x=a if The following conditions must all be fulfilled in order for f to be continuous at x= a : (i) The function f must be defined at x = a . (ii) The limit must exist. (iii) L= f ( a ). lim ( ) x a f x L = Def: The function f(x) is discontinuous in x=a if f(x) is not continuous in x=a. Def: The function f(x) is continuous over the interval [a,b] if f(x) is continuous in every point in [a,b].

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