Midterm II Course: ECO10002 Student: QUESTION 1 1 Let f (.x) = 200x 3 be the output quantity of a firm when the input is x. Suppose each unit of output is sold at a price of 30 euros. The cost function is C(x) = 15x 3 . Find the profit function Tr(x) and find the value of x 2 0 that maximizes profit. 0 0 0 1 0 10 0 20 0 50 0 100 0 200 QUESTION 2 Is the profit function Tr(x) convex (from Question 1)? O True O False O Indefinite QUESTION 3 Find the maximizer of the function y =xe" , where x E [- 1, 5]. A maximizer is a value of x that maximizes the function. 0 -1 0 0 01 02 0 3 0 4 0 5 QUESTION 4 Find the interval of concavity for the following function: f (x) = x + O ( - 00 , + 0 ) 0 ( - 00 , 0] 0 ( - 00, 0) 0 ( - 00, 1) 0 ( - 1, + 00 ) 0 [0, + 00 ) 0 (0, + 00 ) 0 ( - 1, 1 ) QUESTION 5 Find the interval of convexity for the following function: f (x) = (x+2)e-* O ( - 00 , + 00 ) 0 ( - 00 , 0] 0 ( - 00, 0) 0 ( - 00 , 1) 0 ( - 1, + 00 ) 0 10, + 00 ) 0 (0, + 00 ) 0 (-1, 1) QUESTION 6 The price a firm obtains for a commodity varies with demand Q according to the formula P(Q) = 18 - 0.006Q. Total cost is C(Q) = 0.00402 + 4Q + 450 Find the firm's profit IT(Q) and the value of Q* which maximizes profit. Choose the correct Q*. 0 0 100 500 0 700 1000 0 1100 O NA QUESTION 7 Find the maximum point of the function:f (x) = 4x2- 40x + 80 for x E [0.8] (Hint: you need to find a maximizer and then evaluate the function at that value) 1