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A company produces 3 products, A, B, and C, that require processing by three machines, I, II and III. The time in hours that is required for processing one unit of each product by the three machines is given by the following table: ABC 312 II 1 2 1 111 2 4 1 Machine I is available for 440 hours, machine II for 310 hours and machine III for 560 hours. Find how many units of each product should be produced to make use of ALL available time on the machines. Letx, y, and z represent the number of product A, B, and C produced.

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3. A company manufactures two products (A and B) and the profit per unit sold is $3 and $5 respectively. Each product has to be assembled on a particular machine, with each unit of product A taking 12 minutes of assembly time and each unit of product B taking 25 minutes of assembly time. The company estimates that the machine used for assembly has an effective working week of only 30 hours (due to maintenance/breakdown). Technological constraints mean that for every five units of product A produced at least two units of product B must be produced. Which system of inequalities will describe the constraints in this problem, if x, is the number of units of product A produced, and x, is the number of units of product B produced in a week? 25x4 +12x, 51800 12x, +25x, S1800 12x, +25x, s1800 A) 5x, 22x B) 2x, 25x C) 5x + 2x XA20, X20 X20, X, 20 120, X20 25x, +12x, 51800 D) 2x 25x4 E) none of these XA20, X720

A utility function is a function that attaches a measure to the satisfaction or utility a consumer gets from the consumption of products per unit time. Suppose U (x, y, z)= xyz is the utility function for 3 products, A, B, and C, with x, y, and z representing the number of units of 3 different products, respectively. The prices of these three products are pa, pe, and pc, in dollars, given by: PA=1 Pa = 2 Pc =1 Find the values of x, y, and that will maximize U (x, y, z)= xyz, subject to the condition that you must purchase $600 worth of products. In other words, P.X+ Pey+ Pez=600, with x, y, and z>0.

48 f(x) = 12 + (x + 1)2 4. (22 pts) The function f, its derivative f', and its second derivative f" are given by the expressions below. f'(x) = -96(x+1) (12+(x + 1)2) (12+(x + 1)2) (a) Circle the correct statement. f"(x)288(x2 + 2x - 3) i. f is even ii. f is odd. iii. f is neither even nor odd. (b) Give an equation for each horizontal asymptote of f. Write "none" if appropriate. Show your work. Show work for parts (c), (d), (e), and (f) in the space below. (c) List the interval(s) on which f is increasing. Write "none" if appropriate. (d) List the interval(s) on which f is decreasing. Write "none" if appropriate (e) List the x-value(s) at which the local minima of f occur. Write "none" if appropriate (f) List the x-value(s) at which the local maxima of f occur. Write "none" if appropriate. (8) List the interval(s) on which f is concave up. Write "none" if appropriate. (h) List the interval(s) on which f is concave down. Write "none" if appropriate. (i) List the x-coordinate(s) of the inflection points of f. Write "none" if appropriate. () Sketch the graph of y = f(x). Mark the locations of any local extrema and/or inflection points with filled-in circles. Your sketch should be consistent with your answers to (a)-0). -10 -9 -8 -7 -6 -5 -4 4 5 6 7 8 9 10

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