🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Calculus for Business, Economics, Life Sciences, and Social Sciences 13th ## Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen ## Chapter 4 ## Graphing and Optimization ## Educators  ### Problem 1 Inspect the graph of the function to determine whether it is increasing or decreasing on the given interval.$g(x)=|x|$on$(-\infty, 0)$ Dwijendra R. Numerade Educator ### Problem 2 Inspect the graph of the function to determine whether it is increasing or decreasing on the given interval.$m(x)=x^{3}$on$(0, \infty)$ Dwijendra R. Numerade Educator ### Problem 3 Inspect the graph of the function to determine whether it is increasing or decreasing on the given interval.$f(x)=x$on$(-\infty, \infty)$ Dwijendra R. Numerade Educator ### Problem 4 Inspect the graph of the function to determine whether it is increasing or decreasing on the given interval.$k(x)=-x^{2}$on$(0, \infty)$ Dwijendra R. Numerade Educator ### Problem 5 Inspect the graph of the function to determine whether it is increasing or decreasing on the given interval.$p(x)=\sqrt{x}$on$(-\infty, 0)$ Dwijendra R. Numerade Educator ### Problem 6 Inspect the graph of the function to determine whether it is increasing or decreasing on the given interval.$h(x)=x^{2}$on$(-\infty, 0)$ Dwijendra R. Numerade Educator ### Problem 7 Inspect the graph of the function to determine whether it is increasing or decreasing on the given interval.$r(x)=4-\sqrt{x}$on$(0, \infty)$ Dwijendra R. Numerade Educator ### Problem 8 Inspect the graph of the function to determine whether it is increasing or decreasing on the given interval.$g(x)=|x|$on$(0, \infty)$ Dwijendra R. Numerade Educator ### Problem 9 Refer to the following graph of$y=f(x)$Identify the intervals on which$f(x)$is increasing. Dwijendra R. Numerade Educator ### Problem 10 Refer to the following graph of$y=f(x)$Identify the intervals on which$f(x)$is decreasing. Dwijendra R. Numerade Educator ### Problem 11 Refer to the following graph of$y=f(x)$Identify the intervals on which$f^{\prime}(x)<0$ Dwijendra R. Numerade Educator ### Problem 12 Refer to the following graph of$y=f(x)$Identify the intervals on which$f^{\prime}(x)>0 .$ Dwijendra R. Numerade Educator ### Problem 13 Refer to the following graph of$y=f(x)$Identify the$x$coordinates of the points where$f^{\prime}(x)=0$. Dwijendra R. Numerade Educator ### Problem 14 Refer to the following graph of$y=f(x)$Identify the$x$coordinates of the points where$f^{\prime}(x)$does not exist. Dwijendra R. Numerade Educator ### Problem 15 Refer to the following graph of$y=f(x)$Identify the$x$coordinates of the points where$f(x)$has a local maximum. Dwijendra R. Numerade Educator ### Problem 16 Refer to the following graph of$y=f(x)$Identify the$x$coordinates of the points where$f(x)$has a local minimum. Dwijendra R. Numerade Educator ### Problem 17$f(x)$is continuous on$(-\infty, \infty)$and has critical numbers at$x=a, b, c,$and$d$. Use the sign chart for$f^{\prime}(x)$to determine whether$f$has a local maximum, a local minimum, or neither at each critical number. Dwijendra R. Numerade Educator ### Problem 18$f(x)$is continuous on$(-\infty, \infty)$and has critical numbers at$x=a, b, c,$and$d$. Use the sign chart for$f^{\prime}(x)$to determine whether$f$has a local maximum, a local minimum, or neither at each critical number. Dwijendra R. Numerade Educator ### Problem 19 Give the local extrema off and match the graph off with one of the sign charts$a$-$h$in the figure on page$250 .$ Dwijendra R. Numerade Educator ### Problem 20 Give the local extrema off and match the graph off with one of the sign charts$a$-$h$in the figure on page$250 .$ Dwijendra R. Numerade Educator ### Problem 21 Give the local extrema off and match the graph off with one of the sign charts$a$-$h$in the figure on page$250 .$ Dwijendra R. Numerade Educator ### Problem 22 Give the local extrema off and match the graph off with one of the sign charts$a$-$h$in the figure on page$250 .$ Dwijendra R. Numerade Educator ### Problem 23 Give the local extrema off and match the graph off with one of the sign charts$a$-$h$in the figure on page$250 .$ Dwijendra R. Numerade Educator ### Problem 24 Give the local extrema off and match the graph off with one of the sign charts$a$-$h$in the figure on page$250 .$ Dwijendra R. Numerade Educator ### Problem 25 Give the local extrema off and match the graph off with one of the sign charts$a$-$h$in the figure on page$250 .$ Dwijendra R. Numerade Educator ### Problem 26 Give the local extrema off and match the graph off with one of the sign charts$a$-$h$in the figure on page$250 .$ Dwijendra R. Numerade Educator ### Problem 27 Find$\left(\right.$A)$f^{\prime}(x),($B) the partition numbers for$f^{\prime},$and$($C) the critical numbers of$f$.$f(x)=x^{3}-12 x+8$ Dwijendra R. Numerade Educator ### Problem 28 Find$\left(\right.$A)$f^{\prime}(x),($B) the partition numbers for$f^{\prime},$and$($C) the critical numbers of$f$.$f(x)=x^{3}-27 x+30$ Dwijendra R. Numerade Educator ### Problem 29 Find$\left(\right.$A)$f^{\prime}(x),($B) the partition numbers for$f^{\prime},$and$($C) the critical numbers of$f$.$f(x)=\frac{6}{x+2}$ Dwijendra R. Numerade Educator ### Problem 30 Find$\left(\right.$A)$f^{\prime}(x),($B) the partition numbers for$f^{\prime},$and$($C) the critical numbers of$f$.$f(x)=\frac{5}{x-4}$ Dwijendra R. Numerade Educator ### Problem 31 Find$\left(\right.$A)$f^{\prime}(x),($B) the partition numbers for$f^{\prime},$and$($C) the critical numbers of$f$.$f(x)=|x|$ Dwijendra R. Numerade Educator ### Problem 32 Find$\left(\right.$A)$f^{\prime}(x),($B) the partition numbers for$f^{\prime},$and$($C) the critical numbers of$f$.$f(x)=|x+3|$ Dwijendra R. Numerade Educator ### Problem 33 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=2 x^{2}-4 x$ Dwijendra R. Numerade Educator ### Problem 34 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=-3 x^{2}-12 x$ Dwijendra R. Numerade Educator ### Problem 35 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=-2 x^{2}-16 x-25$ Dwijendra R. Numerade Educator ### Problem 36 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=-3 x^{2}+12 x-5$ Dwijendra R. Numerade Educator ### Problem 37 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=x^{3}+4 x-5$ Dwijendra R. Numerade Educator ### Problem 38 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=-x^{3}-4 x+8$ Dwijendra R. Numerade Educator ### Problem 39 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=2 x^{3}-3 x^{2}-36 x$ Dwijendra R. Numerade Educator ### Problem 40 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=-2 x^{3}+3 x^{2}+120 x$ Dwijendra R. Numerade Educator ### Problem 41 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=3 x^{4}-4 x^{3}+5$ Dwijendra R. Numerade Educator ### Problem 42 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=x^{4}+2 x^{3}+5$ Dwijendra R. Numerade Educator ### Problem 43 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=(x-1) e^{-x}$ Dwijendra R. Numerade Educator ### Problem 44 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=x \ln x-x$ Dwijendra R. Numerade Educator ### Problem 45 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=4 x^{1 / 3}-x^{2 / 3}$ Dwijendra R. Numerade Educator ### Problem 46 Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=\left(x^{2}-9\right)^{2 / 3}$ Dwijendra R. Numerade Educator ### Problem 47 Use a graphing calculator to approximate the critical numbers of$f(x)$to two decimal places. Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=x^{4}-4 x^{3}+9 x$ Dwijendra R. Numerade Educator ### Problem 48 Use a graphing calculator to approximate the critical numbers of$f(x)$to two decimal places. Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=x^{4}+5 x^{3}-15 x$ Dwijendra R. Numerade Educator ### Problem 49 Use a graphing calculator to approximate the critical numbers of$f(x)$to two decimal places. Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=x \ln x-(x-2)$Check back soon! ### Problem 50 Use a graphing calculator to approximate the critical numbers of$f(x)$to two decimal places. Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=e^{-x}-3 x^{2}$ Dwijendra R. Numerade Educator ### Problem 51 Use a graphing calculator to approximate the critical numbers of$f(x)$to two decimal places. Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=e^{x}-2 x^{2}$ Dwijendra R. Numerade Educator ### Problem 52 Use a graphing calculator to approximate the critical numbers of$f(x)$to two decimal places. Find the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema.$f(x)=\frac{\ln x}{x}-5 x+x^{2}$ Dwijendra R. Numerade Educator ### Problem 53 Find the intervals on which$f(x)$is increasing and the intervals on which$f(x)$is decreasing. Then sketch the graph. Add horizontal tangent lines.$f(x)=4+8 x-x^{2}$ Dwijendra R. Numerade Educator ### Problem 54 Find the intervals on which$f(x)$is increasing and the intervals on which$f(x)$is decreasing. Then sketch the graph. Add horizontal tangent lines.$f(x)=2 x^{2}-8 x+9$ Dwijendra R. Numerade Educator ### Problem 55 Find the intervals on which$f(x)$is increasing and the intervals on which$f(x)$is decreasing. Then sketch the graph. Add horizontal tangent lines.$f(x)=x^{3}-3 x+1$ Dwijendra R. Numerade Educator ### Problem 56 Find the intervals on which$f(x)$is increasing and the intervals on which$f(x)$is decreasing. Then sketch the graph. Add horizontal tangent lines.$f(x)=x^{3}-12 x+2$ Dwijendra R. Numerade Educator ### Problem 57 Find the intervals on which$f(x)$is increasing and the intervals on which$f(x)$is decreasing. Then sketch the graph. Add horizontal tangent lines.$f(x)=10-12 x+6 x^{2}-x^{3}$ Dwijendra R. Numerade Educator ### Problem 58 Find the intervals on which$f(x)$is increasing and the intervals on which$f(x)$is decreasing. Then sketch the graph. Add horizontal tangent lines.$f(x)=x^{3}+3 x^{2}+3 x$ Dwijendra R. Numerade Educator ### Problem 59 Find the intervals on which$f(x)$is increasing and the intervals on which$f(x)$is decreasing. Then sketch the graph. Add horizontal tangent lines.$f(x)=x^{4}-18 x^{2}$ Dwijendra R. Numerade Educator ### Problem 60 Find the intervals on which$f(x)$is increasing and the intervals on which$f(x)$is decreasing. Then sketch the graph. Add horizontal tangent lines.$f(x)=-x^{4}+50 x^{2}$ Dwijendra R. Numerade Educator ### Problem 61$f(x)$is continuous on$(-\infty, \infty) .$Use the given information to sketch the graph off. Aman G. Numerade Educator ### Problem 62$f(x)$is continuous on$(-\infty, \infty) .$Use the given information to sketch the graph off. Aman G. Numerade Educator ### Problem 63$f(x)$is continuous on$(-\infty, \infty) .$Use the given information to sketch the graph off. Aman G. Numerade Educator ### Problem 64$f(x)$is continuous on$(-\infty, \infty) .$Use the given information to sketch the graph off. Aman G. Numerade Educator ### Problem 65$f(x)$is continuous on$(-\infty, \infty) .$Use the given information to sketch the graph off.$f(-2)=4, f(0)=0, f(2)=-4f^{\prime}(-2)=0, f^{\prime}(0)=0, f^{\prime}(2)=0f^{\prime}(x)>0$on$(-\infty,-2)$and$(2, \infty)f^{\prime}(x)<0$on (-2,0) and (0,2) Aman G. Numerade Educator ### Problem 66$f(x)$is continuous on$(-\infty, \infty) .$Use the given information to sketch the graph off.$f(-2)=-1, f(0)=0, f(2)=1f^{\prime}(-2)=0, f^{\prime}(2)=0f^{\prime}(x)>0$on$(-\infty,-2),(-2,2),$and$(2, \infty)$ Aman G. Numerade Educator ### Problem 67$f(x)$is continuous on$(-\infty, \infty) .$Use the given information to sketch the graph off.$f(-1)=2, f(0)=0, f(1)=-2 ;f^{\prime}(-1)=0, f^{\prime}(1)=0, f^{\prime}(0)$is not defined;$f^{\prime}(x)>0$on$(-\infty,-1)$and$(1, \infty)f^{\prime}(x)<0$on (-1,0) and (0,1) Aman G. Numerade Educator ### Problem 68$f(x)$is continuous on$(-\infty, \infty) .$Use the given information to sketch the graph off.$f(-1)=2, f(0)=0, f(1)=2f^{\prime}(-1)=0, f^{\prime}(1)=0, f^{\prime}(0)$is not defined;$f^{\prime}(x)>0$on$(-\infty,-1)$and (0,1)$f^{\prime}(x)<0$on (-1,0) and$(1, \infty)$ Aman G. Numerade Educator ### Problem 69 Involve functions$f_{1}-f_{6}$and their derivatives,$g_{1}-g_{6} .$Use the graphs shown in figures (A) and ( B) to match each function$f_{i}$with its derivative$g_{j}$.$f_{1}$ Aman G. Numerade Educator ### Problem 70 Involve functions$f_{1}-f_{6}$and their derivatives,$g_{1}-g_{6} .$Use the graphs shown in figures (A) and ( B) to match each function$f_{i}$with its derivative$g_{j}$.$f_{2}$ Aman G. Numerade Educator ### Problem 71 Involve functions$f_{1}-f_{6}$and their derivatives,$g_{1}-g_{6} .$Use the graphs shown in figures (A) and ( B) to match each function$f_{i}$with its derivative$g_{j}$.$f_{3}$ Aman G. Numerade Educator ### Problem 72 Involve functions$f_{1}-f_{6}$and their derivatives,$g_{1}-g_{6} .$Use the graphs shown in figures (A) and ( B) to match each function$f_{i}$with its derivative$g_{j}$.$f_{4}$ Aman G. Numerade Educator ### Problem 73 Involve functions$f_{1}-f_{6}$and their derivatives,$g_{1}-g_{6} .$Use the graphs shown in figures (A) and ( B) to match each function$f_{i}$with its derivative$g_{j}$.$f_{5}$ Aman G. Numerade Educator ### Problem 74 Involve functions$f_{1}-f_{6}$and their derivatives,$g_{1}-g_{6} .$Use the graphs shown in figures (A) and ( B) to match each function$f_{i}$with its derivative$g_{j}$.$f_{6}$ Aman G. Numerade Educator ### Problem 75 Use the given graph of$y=f^{\prime}(x)$to find the intervals on which$f$is increasing, the intervals on which$f$is decreasing, and the x coordinates of the local extrema off. Sketch a possible graph of$y=f(x)$. Aman G. Numerade Educator ### Problem 76 Use the given graph of$y=f^{\prime}(x)$to find the intervals on which$f$is increasing, the intervals on which$f$is decreasing, and the x coordinates of the local extrema off. Sketch a possible graph of$y=f(x)$. Aman G. Numerade Educator ### Problem 77 Use the given graph of$y=f^{\prime}(x)$to find the intervals on which$f$is increasing, the intervals on which$f$is decreasing, and the x coordinates of the local extrema off. Sketch a possible graph of$y=f(x)$. Aman G. Numerade Educator ### Problem 78 Use the given graph of$y=f^{\prime}(x)$to find the intervals on which$f$is increasing, the intervals on which$f$is decreasing, and the x coordinates of the local extrema off. Sketch a possible graph of$y=f(x)$. Aman G. Numerade Educator ### Problem 79 Use the given graph of$y=f^{\prime}(x)$to find the intervals on which$f$is increasing, the intervals on which$f$is decreasing, and the x coordinates of the local extrema off. Sketch a possible graph of$y=f(x)$. Aman G. Numerade Educator ### Problem 80 Use the given graph of$y=f^{\prime}(x)$to find the intervals on which$f$is increasing, the intervals on which$f$is decreasing, and the x coordinates of the local extrema off. Sketch a possible graph of$y=f(x)$. Aman G. Numerade Educator ### Problem 81 Use the given graph of$y=f(x)$to find the intervals on which$f^{\prime}(x)>0,$the intervals on which$f^{\prime}(x)<0,$and the values of$x$for which$f^{\prime}(x)=0 .$Sketch a possible graph of$y=f^{\prime}(x)$. Aman G. Numerade Educator ### Problem 82 Use the given graph of$y=f(x)$to find the intervals on which$f^{\prime}(x)>0,$the intervals on which$f^{\prime}(x)<0,$and the values of$x$for which$f^{\prime}(x)=0 .$Sketch a possible graph of$y=f^{\prime}(x)$. Aman G. Numerade Educator ### Problem 83 Use the given graph of$y=f(x)$to find the intervals on which$f^{\prime}(x)>0,$the intervals on which$f^{\prime}(x)<0,$and the values of$x$for which$f^{\prime}(x)=0 .$Sketch a possible graph of$y=f^{\prime}(x)$. Aman G. Numerade Educator ### Problem 84 Use the given graph of$y=f(x)$to find the intervals on which$f^{\prime}(x)>0,$the intervals on which$f^{\prime}(x)<0,$and the values of$x$for which$f^{\prime}(x)=0 .$Sketch a possible graph of$y=f^{\prime}(x)$. Aman G. Numerade Educator ### Problem 85 Find the critical numbers, the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema. Do not graph.$f(x)=x+\frac{4}{x}$ Aman G. Numerade Educator ### Problem 86 Find the critical numbers, the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema. Do not graph.$f(x)=\frac{9}{x}+x$ Aman G. Numerade Educator ### Problem 87 Find the critical numbers, the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema. Do not graph.$f(x)=1+\frac{1}{x}+\frac{1}{x^{2}}$ Aman G. Numerade Educator ### Problem 88 Find the critical numbers, the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema. Do not graph.$f(x)=3-\frac{4}{x}-\frac{2}{x^{2}}$ Aman G. Numerade Educator ### Problem 89 Find the critical numbers, the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema. Do not graph.$f(x)=\frac{x^{2}}{x-2}$ Aman G. Numerade Educator ### Problem 90 Find the critical numbers, the intervals on which$f(x)$is increasing, the intervals on which$f(x)$is decreasing, and the local extrema. Do not graph.$f(x)=\frac{x^{2}}{x+1}$ Aman G. Numerade Educator ### Problem 91 The graph of the total profit$P(x)$(in dollars) from the sale of$x$cordless electric screwdrivers is shown in the figure. (A) Write a brief description of the graph of the marginal profit function$y=P^{\prime}(x)$, including a discussion of any$x$intercepts. (B) Sketch a possible graph of$y=P^{\prime}(x)$. Aman G. Numerade Educator ### Problem 92 The graph of the total revenue$R(x)$(in dollars) from the sale of$x$cordless electric screwdrivers is shown in the figure. (A) Write a brief description of the graph of the marginal revenue function$y=R^{\prime}(x),$including a discussion of any$x$intercepts. (B) Sketch a possible graph of$y=R^{\prime}(x)$. Aman G. Numerade Educator ### Problem 93 The figure approximates the rate of change of the price of bacon over a 70 -month period, where$B(t)$is the price of a pound of sliced bacon (in dollars) and$t$is time (in months). (A) Write a brief description of the graph of$y=B(t)$, including a discussion of any local extrema. (B) Sketch a possible graph of$y=B(t)$. Aman G. Numerade Educator ### Problem 94 The figure approximates the rate of change of the price of eggs over a 70 -month period, where$E(t)$is the price of a dozen eggs (in dollars) and$t$is time (in months). (A) Write a brief description of the graph of$y=E(t)$, including a discussion of any local extrema. (B) Sketch a possible graph of$y=E(t)$. Aman G. Numerade Educator ### Problem 95 A manufacturer incurs the following costs in producing$x$water ski vests in one day, for$0<x<150$: fixed costs,$\$320 ;$ unit production cost, $\$ 20$per vest; equipment maintenance and repairs,$0.05 x^{2}$dollars. So, the cost of manufacturing$x$vests in one day is given by $$C(x)=0.05 x^{2}+20 x+320 \quad 0<x<150$$ (A) What is the average$\operatorname{cost} \bar{C}(x)$per vest if$x$vests are produced in one day? (B) Find the critical numbers of$\bar{C}(x),$the intervals on which the average cost per vest is decreasing, the intervals on which the average cost per vest is increasing, and the local extrema. Do not graph. Aman G. Numerade Educator ### Problem 96 A manufacturer incurs the following costs in producing$x$rain jackets in one day for$0<x<200$: fixed costs,$\$450 ;$ unit production cost, $\$ 30$per jacket; equipment maintenance and repairs,$0.08 x^{2}$dollars. (A) What is the average cost$\bar{C}(x)$per jacket if$x$jackets are produced in one day? (B) Find the critical numbers of$\bar{C}(x),$the intervals on which the average cost per jacket is decreasing, the intervals on which the average cost per jacket is increasing, and the local extrema. Do not graph. Aman G. Numerade Educator ### Problem 97 A drug is injected into the bloodstream of a patient through the right arm. The drug concentration in the bloodstream of the left arm$t$hours after the injection is approximated by $$C(t)=\frac{0.28 t}{t^{2}+4} \quad 0<t<24$$ Find the critical numbers of$C(t),$the intervals on which the drug concentration is increasing, the intervals on which the concentration of the drug is decreasing, and the local extrema. Do not graph. Aman G. Numerade Educator ### Problem 98 The concentration$C(t),$in milligrams per cubic centimeter, of a particular drug in a patient's bloodstream is given by $$C(t)=\frac{0.3 t}{t^{2}+6 t+9} \quad 0<t<12$$ where$t$is the number of hours after the drug is taken orally. Find the critical numbers of$C(t),\$ the intervals on which the drug concentration is increasing, the intervals on which the drug concentration is decreasing, and the local extrema. Do not graph. Aman G.