4. At exactly two of the labeled points in Figure 2.59, the derivative f is 0; the second derivative f" is not zero at any of the labeled points. On a copy of the table, give the signs of f, f, f" at each marked point. B C A D Figure 2.59 Point A B D6. Sketch the graph of a function whose first derivative is everywhere negative and whose second derivative is positive for some X-values and negative for other x-values. In Problems 19-24, graph the second derivative of the function.\f26. The concentration, N(t), of nitrous oxide (a greenhouse gas) in the air is increasing faster with time, t.24 a. What is the sign of i. N'(t)? ii. N" (t)? b. Which is larger i. N(10) or N(20)? ii. N'(10) or N'(20)?30. In economics, total utility refers to the total satisfaction from consuming some commodity. According to the economist Paul Samuelson:27 As you consume more of the same good, the total (psychological) utility increases. However, . . . with successive new units of the good, your total utility will grow at a slower and slower rate because of a fundamental tendency for your psychological ability to appreciate more of the good to become less keen. a. Sketch the total utility as a function of the number of units consumed. b. In terms of derivatives, what is Samuelson saying?32. An industry is being charged by the Environmental Protection Agency (EPA) with dumping unacceptable levels of toxic pollutants in a lake. Over a period of several months, an engineering firm makes daily measurements of the rate at which pollutants are being discharged into the lake. The engineers produce a graph similar to either Figure 2.60(a) or Figure 2.60(b). For each case, give an idea of what argument the EPA might make in court against the industry and in the industry's defense. (a) rate of discharge (b) rate of discharge time time a year ago now a year ago now Figure 2.6040. Figure 2.62 gives the position, f(t), of a particle at time t. At which of the marked values of t can the following statements be true? a. The position is positive b. The velocity is positive c. The acceleration is positive d. The position is decreasing e. The velocity is decreasing f (t) t 14 15 Figure 2.6244. The graph of f (not f) is given in Figure 2.64. At which of the marked values of x is a. f(x) greatest? b. f(x) least? c. f (x) greatest? d. f (x) least? e. f(x) greatest? f. f' (x) least? f' (x ) X X1 X2 X3 X4 X5 X6 Figure 2.64: Graph of f, not fAns t Let us consider f (x) = - 23+ 3x2- 4x 8 ( 20 ) = - 3x2 + 621 - 4 Now the coefficient of 212 is -we indicating a downward facing parabola . Also the coefficient of a is the - ? the graph of f(x ) is decreasing in values fix) is - we x values Now to prove f''(x) is the for some a values and - we for others. = ) 8 " ( x ) = - 6x + 6 Now the coefficient of re is - we indicating a downward facing linear function.' since the cafficient is-we of"(7) is -we for a ? 1 and $" () isthe fox x