problems3

# problems3 - MATH 285-3 SUPPLEMENTAL HOMEWORK PROBLEMS 4.2...

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MATH 285-3 SUPPLEMENTAL HOMEWORK PROBLEMS 4.2 Extrema 1 Find all critical points and determine whether they are local maxima, local minima, or neither, for the following functions: (i) f ( x , y , z ) = x 2 - 4 y 2 + z 2 + 8 yz + 8 xy , and (ii) f ( x , y , z ) = x 2 + y 2 - z 3 + 3 z . (iii) f ( x , y , z ) = x 4 + x 2 - 6 xy + 3 y 2 + z 2 . 2 A ﬁrm has a monopoly on two products and can set the prices. Assume the demand for the products by consumers is Q 1 = 145 - 2 P 1 - P 2 and Q 2 = 160 - P 1 - 3 P 2 . Assume that the cost is given by C = 5 Q 1 + 5 Q 2 , and the proﬁt is given by π = P 1 Q 1 + P 2 Q 2 - C or π = P 1 ( 145 - 2 P 1 - P 2 ) + P 2 ( 160 - P 1 - 3 P 2 ) - 5 ( 145 - 2 P 1 - P 2 ) - 5 ( 160 - P 1 - 3 P 2 ). What are the prices that maximize proﬁt? Use the second derivative test to show that these prices give a point that locally maximizes the proﬁt. 3 A company operates two plants which manufacture the same item and whose cost functions are C 1 = 8 . 5 + 0 . 03 Q 2 1 and C 2 = 5 . 2 + 0 . 04 Q 2 2 , where Q 1 and Q 2 are the quantities produced by each plant, and Q = Q 1 + Q 2 is the total quantity produced. The prices are determined by the total amount produced by P = 60 - 0 . 04 Q . How much should each plant produce in order to maximize the company’s proﬁt? Hint: The proﬁt is π = P Q - C 1 - C 2 = ( 60 - 0 . 04 Q ) Q - C 1 - C 2 = 60 Q - 0 . 04 Q 2 - C 1 - C 2 = 60 ( Q 1 + Q 2 ) - 0 . 04 ( Q 1 + Q 2 ) 2 - 8 . 5 - 0 . 03 Q 2 1 - 5 . 2 - 0 . 04 Q 2 2 .

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## This note was uploaded on 04/07/2008 for the course MATH 285-3 taught by Professor Rogerson during the Spring '08 term at Northwestern.

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problems3 - MATH 285-3 SUPPLEMENTAL HOMEWORK PROBLEMS 4.2...

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