Increasing and Decreasing Functions and the First Derivative Test. Increasing and Decreasing Functions. Video

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1 SECTION and Decreasing Functions and the First Derivative Test 79 Section and Decreasing Functions and the First Derivative Test Determine intervals on which a unction is increasing or decreasing Appl the First Derivative Test to ind relative etrema o a unction and Decreasing Functions In this section ou will learn how derivatives can be used to classi relative etrema as either relative minima or relative maima First, it is important to deine increasing and decreasing unctions Deinitions o and Decreasing Functions Decreasing = a = b A unction is increasing on an interval i or an two numbers and in the interval, < implies < A unction is decreasing on an interval i or an two numbers and in the interval, < implies > Video Constant () < 0 () = 0 () > 0 The derivative is related to the slope o a unction Figure A unction is increasing i, as moves to the right, its graph moves up, and is decreasing i its graph moves down For eample, the unction in Figure is decreasing on the interval, a, is constant on the interval a, b, and is increasing on the interval b, As shown in Theorem below, a positive derivative implies that the unction is increasing; a negative derivative implies that the unction is decreasing; and a zero derivative on an entire interval implies that the unction is constant on that interval THEOREM Test or and Decreasing Functions Let be a unction that is continuous on the closed interval a, b and dierentiable on the open interval a, b I > 0 or all in a, b, then is increasing on a, b I < 0 or all in a, b, then is decreasing on a, b I 0 or all in a, b, then is constant on a, b Proo To prove the irst case, assume that > 0 or all in the interval a, b and let < be an two points in the interval B the Mean Value Theorem, ou know that there eists a number c such that < c <, and c Because c > 0 and > 0, ou know that > 0 which implies that < So, is increasing on the interval The second case has a similar proo (see Eercise 0), and the third case was given as Eercise 78 in Section NOTE The conclusions in the irst two cases o Theorem are valid even i 0 at a inite number o -values in a, b

2 80 CHAPTER Applications o Dierentiation EXAMPLE Intervals on Which Is or Decreasing Find the open intervals on which is increasing or decreasing () = Solution Note that is dierentiable on the entire real number line To determine the critical numbers o, set equal to zero Write original unction (0, 0) (, ) Figure Editable Graph Decreasing 0 Dierentiate and set equal to 0 0 Factor 0, Critical numbers Because there are no points or which does not eist, ou can conclude that 0 and are the onl critical numbers The table summarizes the testing o the three intervals determined b these two critical numbers Interval < < 0 Test Value Sign o > 0 0 < < < 0 < < > 0 Conclusion Decreasing (a) Strictl monotonic unction (b) Not strictl monotonic Figure 7 () = Constant () =, 0, ( ), < 0 0 > So, is increasing on the intervals, 0 and, and decreasing on the interval 0,, as shown in Figure Tr It Eploration A Eploration B Video Eample gives ou one eample o how to ind intervals on which a unction is increasing or decreasing The guidelines below summarize the steps ollowed in the eample Guidelines or Finding Intervals on Which a Function Is or Decreasing Let be continuous on the interval a, b To ind the open intervals on which is increasing or decreasing, use the ollowing steps Locate the critical numbers o in a, b, and use these numbers to determine test intervals Determine the sign o at one test value in each o the intervals Use Theorem to determine whether is increasing or decreasing on each interval These guidelines are also valid i the interval a, b is replaced b an interval o the orm, b, a,, or, A unction is strictl monotonic on an interval i it is either increasing on the entire interval or decreasing on the entire interval For instance, the unction is strictl monotonic on the entire real line because it is increasing on the entire real line, as shown in Figure 7(a) The unction shown in Figure 7(b) is not strictl monotonic on the entire real line because it is constant on the interval 0,

3 SECTION and Decreasing Functions and the First Derivative Test 8 () = The First Derivative Test Ater ou have determined the intervals on which a unction is increasing or decreasing, it is not diicult to locate the relative etrema o the unction For instance, in Figure 8 (rom Eample ), the unction (0, 0) etrema o Figure 8 maimum, minimum ( ) has a relative maimum at the point 0, 0 because is increasing immediatel to the let o 0 and decreasing immediatel to the right o 0 Similarl, has a relative minimum at the point, because is decreasing immediatel to the let o and increasing immediatel to the right o The ollowing theorem, called the First Derivative Test, makes this more eplicit THEOREM The First Derivative Test Let c be a critical number o a unction that is continuous on an open interval I containing c I is dierentiable on the interval, ecept possibl at c, then c can be classiied as ollows I changes rom negative to positive at c, then has a relative minimum at c, c I changes rom positive to negative at c, then has a relative maimum at c, c I is positive on both sides o c or negative on both sides o c, then c is neither a relative minimum nor a relative maimum (+) ( ) ( ) (+) () < 0 () > 0 a c b minimum () > 0 () < 0 a c b maimum (+) (+) ( ) ( ) () > 0 () > 0 a c b () < 0 () < 0 a c b Neither relative minimum nor relative maimum Proo Assume that changes rom negative to positive at c Then there eist a and b in I such that < 0 or all in a, c and > 0 or all in c, b B Theorem, is decreasing on a, c and increasing on c, b So, c is a minimum o on the open interval a, b and, consequentl, a relative minimum o This proves the irst case o the theorem The second case can be proved in a similar wa (see Eercise 0) Video

4 8 CHAPTER Applications o Dierentiation EXAMPLE Appling the First Derivative Test Find the relative etrema o the unction sin in the interval 0, Solution Note that is continuous on the interval 0, To determine the critical numbers o in this interval, set equal to 0 cos 0 Set equal to 0 cos, Critical numbers Because there are no points or which does not eist, ou can conclude that and are the onl critical numbers The table summarizes the testing o the three intervals determined b these two critical numbers Interval 0 < < < < < < () = sin maimum Test Value Sign o < 0 > < 0 Conclusion Decreasing Decreasing minimum π π π π A relative minimum occurs where changes rom decreasing to increasing, and a relative maimum occurs where changes rom increasing to decreasing Figure 9 has a relative mini- B appling the First Derivative Test, ou can conclude that mum at the point where and a relative maimum at the point where as shown in Figure 9 -value where relative minimum occurs -value where relative maimum occurs Editable Graph Tr It Eploration A Eploration B EXPLORATION Comparing Graphical and Analtic Approaches From Section, ou know that, b itsel, a graphing utilit can give misleading inormation about the relative etrema o a graph Used in conjunction with an analtic approach, however, a graphing utilit can provide a good wa to reinorce our conclusions Use a graphing utilit to graph the unction in Eample Then use the zoom and trace eatures to estimate the relative etrema How close are our graphical approimations? Note that in Eamples and the given unctions are dierentiable on the entire real line For such unctions, the onl critical numbers are those or which 0 Eample concerns a unction that has two tpes o critical numbers those or which 0 and those or which is not dierentiable

5 SECTION and Decreasing Functions and the First Derivative Test 8 EXAMPLE Appling the First Derivative Test Find the relative etrema o Solution Begin b noting that is continuous on the entire real line The derivative o General Power Rule () = ( ) / 7 maimum 0, ( ) (, 0) (, 0) minimum minimum You can appl the First Derivative Test to ind relative etrema Figure 0 Simpli is 0 when 0 and does not eist when ± So, the critical numbers are, 0, and The table summarizes the testing o the our intervals determined b these three critical numbers Interval < < < < 0 0 < < < < Test Value Sign o < 0 > 0 < 0 > 0 Conclusion Decreasing Decreasing B appling the First Derivative Test, ou can conclude that has a relative minimum at the point, 0, a relative maimum at the point 0,, and another relative minimum at the point, 0, as shown in Figure 0 Editable Graph Tr It Eploration A TECHNOLOGY PITFALL When using a graphing utilit to graph a unction involving radicals or rational eponents, be sure ou understand the wa the utilit evaluates radical epressions For instance, even though and g are the same algebraicall, some graphing utilities distinguish between these two unctions Which o the graphs shown in Figure is incorrect? Wh did the graphing utilit produce an incorrect graph? () = ( ) / g() = [( ) ] / Which graph is incorrect? Figure

6 8 CHAPTER Applications o Dierentiation When using the First Derivative Test, be sure to consider the domain o the unction For instance, in the net eample, the unction is not deined when 0 This -value must be used with the critical numbers to determine the test intervals EXAMPLE Appling the First Derivative Test Find the relative etrema o Solution Rewrite original unction Dierentiate Rewrite with positive eponent () = + (, ) (, ) minimum minimum -values that are not in the domain o, as well as critical numbers, determine test intervals or Figure Simpli Factor So, is zero at ± Moreover, because 0 is not in the domain o, ou should use this -value along with the critical numbers to determine the test intervals ± Critical numbers, ± is not in the domain o The table summarizes the testing o the our intervals determined b these three -values Interval < < < < 0 0 < < < < Test Value Sign o < 0 > 0 < 0 > 0 Conclusion Decreasing Decreasing B appling the First Derivative Test, ou can conclude that has one relative minimum at the point, and another at the point,, as shown in Figure Editable Graph Tr It Eploration A Open Eploration TECHNOLOGY The most diicult step in appling the First Derivative Test is inding the values or which the derivative is equal to 0 For instance, the values o or which the derivative o is equal to zero are 0 and ± I ou have access to technolog that can perorm smbolic dierentiation and solve equations, use it to appl the First Derivative Test to this unction

7 SECTION and Decreasing Functions and the First Derivative Test 8 I a projectile is propelled rom ground level and air resistance is neglected, the object will travel arthest with an initial angle o I, however, the projectile is propelled rom a point above ground level, the angle that ields a maimum horizontal distance is not (see Eample ) EXAMPLE The Path o a Projectile Neglecting air resistance, the path o a projectile that is propelled at an angle g sec v 0 tan h, where is the height, is the horizontal distance, g is the acceleration due to gravit, v 0 is the initial velocit, and h is the initial height (This equation is derived in Section ) Let g eet per second per second, v 0 eet per second, and h 9 eet What value o will produce a maimum horizontal distance? 0 is Solution To ind the distance the projectile travels, let 0, and use the Quadratic Formula to solve or g sec v 0 tan h 0 sec tan 9 0 sec tan 9 0 At this point, ou need to ind the value o that produces a maimum value o Appling the First Derivative Test b hand would be ver tedious Using technolog to solve the equation however, eliminates most o the mess computations The result is that the maimum value o occurs when radian, or This conclusion is reinorced b sketching the path o the projectile or dierent values o, as shown in Figure O the three paths shown, note that the distance traveled is greatest or 08 dd 0, tan ± tan sec sec 8 8 cos sin sin, 0 0 h = 9 θ = θ = θ = 0 0 The path o a projectile with initial angle Figure Simulation Tr It

8 8 CHAPTER Applications o Dierentiation Eercises or Section The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution o the eercise to print an enlarged cop o the graph In Eercises and, use the graph o to ind (a) the largest open interval on which is increasing, and (b) the largest open interval on which is decreasing In Eercises, identi the open intervals on which the unction is increasing or decreasing 7 sin, 0 < < π 9 0 g 8 h 7 cos, π π π 0 < < h cos 0 < <, π π π In Eercises 7 8, (a) ind the critical numbers o (i an), (b) ind the open interval(s) on which the unction is increasing or decreasing, (c) appl the First Derivative Test to identi all relative etrema, and (d) use a graphing utilit to conirm our results In Eercises 9, consider the unction on the interval 0, For each unction, (a) ind the open interval(s) on which the unction is increasing or decreasing, (b) appl the First Derivative Test to identi all relative etrema, and (c) use a graphing utilit to conirm our results 9 cos 0 sin cos sin cos sin cos sin cos sin sin sin cos In Eercises 7, (a) use a computer algebra sstem to dierentiate the unction, (b) sketch the graphs o and on the same set o coordinate aes over the given interval, (c) ind the critical numbers o in the open interval, and (d) ind the interval(s) on which is positive and the interval(s) on which it is negative Compare the behavior o and the sign o 7 9,, 8 0, 9 t t 0, 0 cos sin t,, cos cos, sin, 0, sin cos, 0, 0 < < 0, 0,

9 SECTION and Decreasing Functions and the First Derivative Test 87 In Eercises and, use smmetr, etrema, and zeros to sketch the graph o How do the unctions and g dier?, t cos t sin t, Think About It In Eercises 0, the graph o is shown in the igure Sketch a graph o the derivative o To print an enlarged cop o the graph, select the MathGraph button In Eercises, use the graph o to (a) identi the interval(s) on which is increasing or decreasing, and (b) estimate the values o at which has a relative maimum or minimum gt sin t g ) 8 Writing About Concepts In Eercises 70, assume that is dierentiable or all The signs o are as ollows > 0 on, < 0 on, > 0 on, Suppl the appropriate inequalit or the indicated value o c Function 7 Think About It The unction is dierentiable on the interval, The table shows the values o or selected values o Sketch the graph o, approimate the critical numbers, and identi the relative etrema g g 7 g 8 g 9 g 0 70 g 0 7 Sketch the graph o the arbitrar unction such that > 0, undeined, < 0, 0 < > 7 A dierentiable unction has one critical number at Identi the relative etrema o at the critical number i and 07 Sign o gc g0 0 g 0 g 0 g0 0 g0 0 g Think About It The unction is dierentiable on the interval 0, The table shows the values o or selected values o Sketch the graph o, approimate the critical numbers, and identi the relative etrema

10 88 CHAPTER Applications o Dierentiation 7 Rolling a Ball Bearing A ball bearing is placed on an inclined plane and begins to roll The angle o elevation o the plane is The distance (in meters) the ball bearing rolls in t seconds is st 9sin t (a) Determine the speed o the ball bearing ater t seconds (b) Complete the table and use it to determine the value o that produces the maimum speed at a particular time 7 Numerical, Graphical, and Analtic Analsis The concentration C o a chemical in the bloodstream t hours ater injection into muscle tissue is C(t) st t 7 t, (a) Complete the table and use it to approimate the time when the concentration is greatest t Ct (b) Use a graphing utilit to graph the concentration unction and use the graph to approimate the time when the concentration is greatest (c) Use calculus to determine analticall the time when the concentration is greatest 77 Numerical, Graphical, and Analtic Analsis Consider the unctions and g sin on the interval 0, (a) Complete the table and make a conjecture about which is the greater unction on the interval 0, g t 0 (b) Use a graphing utilit to graph the unctions and use the graphs to make a conjecture about which is the greater unction on the interval 0, (c) Prove that > g on the interval 0, [Hint: Show that h > 0 where h g ] 78 Numerical, Graphical, and Analtic Analsis Consider the unctions and g tan on the interval 0, (a) Complete the table and make a conjecture about which is the greater unction on the interval 0, g (b) Use a graphing utilit to graph the unctions and use the graphs to make a conjecture about which is the greater unction on the interval 0, (c) Prove that < g on the interval 0, [Hint: Show that h > 0, where h g ] 79 Trachea Contraction Coughing orces the trachea (windpipe) to contract, which aects the velocit v o the air passing through the trachea The velocit o the air during coughing is v kr rr, where k is constant, R is the normal radius o the trachea, and r is the radius during coughing What radius will produce the maimum air velocit? 80 Proit The proit P (in dollars) made b a ast-ood restaurant selling hamburgers is P Find the open intervals on which P is increasing or decreasing 8 Power The electric power P in watts in a direct-current circuit with two resistors and connected in parallel is P vr R R R 0 r < R 000, 0,000 R R where v is the voltage I v and R are held constant, what resistance R produces maimum power? 8 Electrical Resistance The resistance R o a certain tpe o resistor is R 000T T 00 0,000 where R is measured in ohms and the temperature T is measured in degrees Celsius (a) Use a computer algebra sstem to ind drdt and the critical number o the unction Determine the minimum resistance or this tpe o resistor (b) Use a graphing utilit to graph the unction R and use the graph to approimate the minimum resistance or this tpe o resistor 8 Modeling Data The end-o-ear assets or the Medicare Hospital Insurance Trust Fund (in billions o dollars) or the ears 99 through 00 are shown 99: 0; 99: 9; 997: ; 998: 0; 999: ; 000: 77; 00: 087 (Source: US Centers or Medicare and Medicaid Services) (a) Use the regression capabilities o a graphing utilit to ind a model o the orm M at bt c or the data (Let t represent 99) (b) Use a graphing utilit to plot the data and graph the model (c) Analticall ind the minimum o the model and compare the result with the actual data

11 SECTION and Decreasing Functions and the First Derivative Test 89 8 Modeling Data The number o bankruptcies (in thousands) or the ears 988 through 00 are shown 988: 9; 989: 0; 990: 7; 99: 880; 99: 97; 99: 987; 99: 8; 99: 88; 99: 0; 997: 70; 998: 9; 999: 90; 000: 770; 00: 8 (Source: Administrative Oice o the US Courts) (a) Use the regression capabilities o a graphing utilit to ind a model o the orm B at bt ct dt e or the data (Let t 8 represent 988) (b) Use a graphing utilit to plot the data and graph the model (c) Find the maimum o the model and compare the result with the actual data Motion Along a Line In Eercises 8 88, the unction st describes the motion o a particle moving along a line For each unction, (a) ind the velocit unction o the particle at an time t 0, (b) identi the time interval(s) when the particle is moving in a positive direction, (c) identi the time interval(s) when the particle is moving in a negative direction, and (d) identi the time(s) when the particle changes its direction 8 st t t 8 st t 7t 0 87 st t t t 88 st t 0t 8t 80 Motion Along a Line In Eercise 89 and 90, the graph shows the position o a particle moving along a line Describe how the particle s position changes with respect to time 9 minima: 0, 0,, 0 maimum:, 9 minimum:, maima:,,, True or False? In Eercises 9 00, determine whether the statement is true or alse I it is alse, eplain wh or give an eample that shows it is alse 9 The sum o two increasing unctions is increasing 9 The product o two increasing unctions is increasing 97 Ever nth-degree polnomial has n critical numbers 98 An nth-degree polnomial has at most n critical numbers 99 There is a relative maimum or minimum at each critical number 00 The relative maima o the unction are and 0 So, has at least one minimum or some in the interval, 0 Prove the second case o Theorem 0 Prove the second case o Theorem 0 Let > 0 and n > be real numbers Prove that n > n 0 Use the deinitions o increasing and decreasing unctions to prove that is increasing on, 0 Use the deinitions o increasing and decreasing unctions to prove that is decreasing on 0, 89 s t s 9 8 t Creating Polnomial Functions polnomial unction In Eercises 9 9, ind a a n n a n n a a a 0 that has onl the speciied etrema (a) Determine the minimum degree o the unction and give the criteria ou used in determining the degree (b) Using the act that the coordinates o the etrema are solution points o the unction, and that the -coordinates are critical numbers, determine a sstem o linear equations whose solution ields the coeicients o the required unction (c) Use a graphing utilit to solve the sstem o equations and determine the unction (d) Use a graphing utilit to conirm our result graphicall 9 minimum: 0, 0; maimum:, 9 minimum: 0, 0; maimum:, 000