Homework 0 Answers

Homework 0 Answers - 1 = 5 and x 2 = 10 . See f gure 2. 1...

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Economics 111: Intermediate Microeconomics Spring 2008 Homework 0 Answer Key Professor: Pedro Dal Bó 1. Find the derivative for the following functions ( a , b and c are real numbers): a. f ( x )= a + bx f 0 ( x )= b b. f ( x )= a + bx c f 0 ( x )= cbx c 1 c. f ( x )=ln xf 0 ( x )= 1 x d. f ( x )=ln( x a ) f 0 ( x )= a x e. f ( x )= bx ln xf 0 ( x )= b ln x + b f. f ( x )= bx ln x f 0 ( x )= b ln x b (ln x ) 2 2. Find the value of x that maximizes the following function: f ( x )= x x 2 (remem- ber to f nd the f rst and second order conditions). FOC f 0 ( x )=1 2 x =0 . Then, x = 1 2 . SOC f 00 ( x )= 2 < 0 then, it is a maximum. See f gure 1. 3. Find the value of x 1 and x 2 that maximizes the following function: f ( x 1 ,x 2 )=10 x 1 x 2 1 +20 x 2 x 2 2 . FOC: df ( x 1 ,x 2 ) dx 1 =10 2 x 1 =0 df ( x 1 ,x 2 ) dx 2 =20 2 x 2 =0 Solving the equations we obtain that x
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Unformatted text preview: 1 = 5 and x 2 = 10 . See f gure 2. 1 4. Find the value of x 1 and x 2 that maximizes the following function: f ( x 1 , x 2 ) = x 1 x 2 x 2 1 x 2 2 . FOC df ( x 1 ,x 2 ) dx 1 = x 2 2 x 1 = 0 df ( x 1 ,x 2 ) dx 2 = x 1 2 x 2 = 0 Solving the system of equations we have that x 1 = 0 and x 2 = 0 . See f gure 3. 2 EC 111: Some Graphs for Homework 0 January 25, 2008 Figure 1: f ( x ) = x-x 2 1 Figure 2: f ( x,y ) = 10 x-x 2 + 20 y-y 2 2 Figure 3: f ( x,y ) = xy-x 2-y 2 3...
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Homework 0 Answers - 1 = 5 and x 2 = 10 . See f gure 2. 1...

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