Question

Multiple choice questions continued. Please circle the letter of your answer or write the letter of your answer clearly in the space provided. MC 7. (8 pts) Let f(1) = (1-3) Which of the following intervals describes ALL the points where f is concave up? (A) (0,3) (B) (-2,5) (C) (-2,00) (D) (-5,5) (E) (5,00) Answer: MC 8. (8 pts) Consider two numbers whose difference is -40. What is the smallest possible value of their product? (A) 41 (B) 400 (C) - 400 (D) 39 (E) None of the above Answer:

Answer 1

7. We know that a function y = f(x) es concave up, if the graph of its derivative f(x) is increasing nie.. in that case - f"(x) >O Here 20e have given the function, - f(x) = (x-3) ex i Differentiating (2) was, to a we get, f'(x) = -(x-3)ě? +ē? = ē*(1-x+3). op, f'(x) = ēX CH-x) --> (10) Differentiating (23) 20.n.to a we get, f"(X) = -ē (4-2) - ē? = - ^(--x +1). op, f"(x) = - ēX (5-2). Now f will be concave rep if f"(x) > 0 i en zie.gif - ē*(5-2) >0 zie, if 5-x) co spodē? 20 7x7 2.2., if >5 o the required interval for f to be concave zep is (5,00). Opleon (E) is foue. 8. Let the two members be x and y such that 2-Y = - 40, zce, Y = (x+40) -- ☺ Now the product of the above two members is P = xy = a (x+HO) = (2? +40%) = $(a) Day) Now, O'(x) = (2x +HO) "(x) = 2. For maximum on minimum value of 0 we have, Ø'(x) = 0 = 2(x+20) = 0 = U=-20 Now $"(-20) = 20 So at x=-20, 0 has a minimum value. and Pamin = $(-20) = 620)2 +40X(-20) = 100 - 800 - - 400 : o the smallest possible value of their posoduct is (H00).