Pset5Answers - Ec115 Problem Set 5 [One-variable calculus]...

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Ec115 Problem Set 5 [One-variable calculus] (remember to check the second order condition) and sketch a graph of the func- tion: (a) y = x 2 6 x + 3 dy dx = 2 x 6 dy dx = 0 ) 2 x 6 = 0 ) x = 3 d 2 y dx 2 = 2 > 0 ) a minimum at x = 3 where y = 9 18 + 3 = 6 : (b) y = 2 x 2 + 4 x 6 dy dx = 4 x + 4 dy dx = 0 4 x + 4 = 0 ) x = 1 d 2 y dx 2 = 4 < 0 ) a maximum at x = 1 where y = 2 + 4 6 = 4 : (c) y = 3 x 3 27 x 2 + 45 x + 15 dy dx = 9 x 2 54 x + 45 dy dx = 0 ) 9 x 2 54 x + 45 = 0 ) x 2 6 x + 5 = 0 ) ( x 5)( x 1) = 0 ) x = 1 ; 5 : d 2 y dx 2 = 18 x 54 : at x = 1 we have d 2 y dx 2 = 18 54 < 0 ) a maximum at x = 1 where y = 36 at x = 5 we have d 2 y dx 2 = 90 54 > 0 ) a minimum at x = 5 where y =?? : (d) y = x 1+ x 2 : 1
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[quotient rule] dy dx = (1 + x 2 )(1) x (2 x ) (1 + x 2 ) 2 = (1 + x 2 ) 2 x 2 (1 + x 2 ) 2 = 1 x 2 (1 + x 2 ) 2 dy dx = 0 ) 1 x 2 (1 + x 2 ) 2 = 0 ) 1 x 2 = 0 ) x = 1 and -1. [quotient rule] d 2 y dx 2 = (1 + x 2 ) 2 ( 2 x ) (1 x 2 )2(1 + x 2 )(2 x ) (1 + x 2 ) 4 [take out common factor] = (1 + x 2 )( 2 x ) (1 x 2 )2(2 x ) (1 + x 2 ) 3 [expand brackets] = 2 x 2 x 3 4 x + 4 x 3 (1 + x 2 ) 3 [tidy up] = 6 x + 2 x 3 (1 + x 2 ) 3 = 2 x ( x 2 3) (1 + x 2 ) 3 : at x = 1 ; d 2 y dx 2 = 2(1 3) 2 3 > 0 ) a minimum at x = 1 (and y = 1 2 ) at x = 1 ; d 2 y dx 2 = 2(1 3) 2 3 < 0 ) a maximum at x = 1 (and y = 1 2 ) : 2. (a) A monopolist faces (inverse) demand function p = 100 q where p is the price of the good and q is the quantity demanded. Suppose the unit cost of production is 10; i.e. the cost of producing output
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This note was uploaded on 12/31/2011 for the course ECON MR 102 taught by Professor Huyduong during the Winter '11 term at RMIT Vietnam.

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Pset5Answers - Ec115 Problem Set 5 [One-variable calculus]...

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