CALCULUSTHOMAS'CALCULUSARLYTRANSCENDChapter 4JAMESSTEWARTApplications of DerivativesAAStCiHA4.11.Absolute(Global)ExtremeValuesExtreme Values of FunctionsSlide 4-4ofDEFINITIONSLet f be a finction with domain D. Then f has an absolute (orLocal maximuglobal) maximum valuc on D at a point c ifNogresocatintf(x) ≤ f(c)for allx in Drvapoffnearbyand an absolute (or global) minimum value on D at c irAbsolute minimu(x)≥ (c)for all x in DNosmallervalueofLocal minimunFanywhere,AlboaNo smaler value oflocal minimmnearbyFIGURE 4.5How to classify maxima and minimaSlide 4- 5Slide 4-
2016/11/15 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 4 Applications of Derivatives Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4.1 Extreme Values of Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 4 1. Absolute (Global) Extreme Values Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 6
Exploring Extreme ValuesFunction ruleDomain DAbsolute extrema on Dmsin(a)y=r(00,00)No absolute maximum.Absolute minimum of 0 at x = 0.(b) y=x2[0, 2]Absolute maximum of 4 at x = 2.Absoluteminimum of 0 atx=0(c) y = x2(0, 2)Absolute maximum of 4 at x = 2.No absolute minimum(d) y = 2(0, 2)No absolute extrema.FIGURE4.1AbsoluteextremaforFunctionswiththesamedefiningrulecanhavedifferentthe sine and cosine functions onextrema,dependingonthedomain.[w/2, w/2]. These values can dependon the domain of a function.lide 4-7Slide4-8=107THEOREM1The Extreme Value Theorem(a) abs min cely(b) abs max and minIf f isContinuougon aClosed intervaD(a,b],then J attains both an absolute max-imum value M and an absolute minimum value m in [a, b]. That is, there arenumbers , and xg in [a, b] with f(x,) = m, f(x) M, and m ≤ f(x) ≤ M forevery other.x in [a, b] (Figure 4.3)The requirements in Theorem 1 that the interval be closed andfinite,and the functionconbinuous arekeyingredients.(c) abs mux oely(d) nomax or minFIGURE4.2Graphs for Example 1alide 4-9Slide 4- 10"Fu)E shows the requirements in Theorem-1that the interval be closed and trefunctioncontinuous arekeyingredientsWithoutthem,the conctusion oftheNo largest valuetheoremneed not hold.OFIGURE4.4Even a single point ofy=xdiscontinuity cankeepafunctionfrom0≤x<1having either a maximum or minimumvalue on a closed interval, The function0Smallest value0≤x<:1Mainrn'siMinismm at intorior poiris continuous at every point of[0, 1]except x = 1. yet its graph over [0, ]]FIGURE 4.3Some possibilities fora continusfiunction'smaximumanddoes not have a highest point.minimum ona csed ierval [a,Slide 4.12Slide 4-11
2016/11/15 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 7 Exploring Extreme Values Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 8 Functions with the same defining rule can have different extrema, depending on the domain. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 10 The requirements in Theorem 1 that the interval be closed and finite, and the function continuous are key ingredients. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 12 It shows the requirements in Theorem 1 that the interval be closed and the function continuous are key ingredients. Without them, the conclusion of the theorem need not hold
ExerciseExercisesFinding Extrema from Graphs3In Exercises 1-6, determine from the graph whether the function hasay absolute extreme values on [o, b], Thexplain how your answris consistent with Theorem 1.3. No absolute minimum, An absolute maximum at x = c.Since the function's domain is an open intervsl, tbeextreme values because f is continuous on [a,b]L. An absolute minimum at x = Co an absolute maximum at x = b.4, No abelute extrema.The function is neither costisuossTheorem I guarantees the existence of suchBor defined ona closed interval, soit neednot fullextreme values because h is continuous on [a,b].the conclusions of Theorem 1.2. An absolute minimum at x = b, an absolute maximum at x =c.Theorem I guarantees the existence of suchextremaluesbcausescontinuouson [b]Slide4-13Slide 4-14In Exerczses 7-10, find the absolute extreme values and where they occurExercisesExercises2R5. An sbeolute minimum at x = a and an absolute maximum at x = e.num st (1,0), local maximum at (1,0)7. Local miNote that y = g(x) is mot continuous butstil has extrema.When the hypothesis of Theorem18. Minima at (2,0) and (2,0), maximum at (0,2)is satisfied then extrema are guaranteed, but when thehypotbesis is aot satiafied, absolute extrema may or may not occuz.6. Absolute minimum at x = c and an abeolute maximum at x = aNote that y=g(x) is not continuous but stilhas absolute extrema, When the hypothesisof Theorem lissatiafied then extrema sre guaranteed, but whenthe hypothesis is not satisfied, absolute extrema may ot may not occurSlide 4-15Slide 4- 18ExercisesExercisesf'r'(t)doesdoes tnot exist9. Maximum at (0,5). Note that there is no minimusince the endpoint (2,0) is excluded from the graph.10, Local maximum at (3,0), local minimum st (2,0),maximum at (1,2), minimum at (0, 1)Slide 4-17
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DEFINITIONAn interior point of the domain ofa function J where is zero orundefined is a critical point of f.2.Method forFinding AbsoluteExtreme Values on a ClosedHow to Find the Absolute Extrema ofa Continuous FunctionfonaFinite Closed IntervalInterval1. Evaluate f at all critical points and endpoints.2. Take the largest and smallest of these values.lide 4-19Slide 4- 20EXAMPLE2Find the absolute maximum and minimum values of j(x) = x on[2, 1].Solution The function is differentiable over its entire domain, so the only critical point isCricalPointswhere J'(x) Zx 0, namely x 0. We need to check the function's values at x = (Need Not Giveand at the endpoints x =2 and x = 1:Extreme ValuesCritical point value: J(0) = 0J(2) = 4Endpoint values:Lf(1) -1The function hasan absolute maximum value of 4 at x-2 and an absolute minimuvalue of 0 atx 0FIGURE 4.7Critical pointswithouextreme values. (a) y = 3r is 0 atx = 0.but y = x has no extremum there.(b) y = (1/3)x-/ is undefined at x = 0,but y =x//has no extremum thereSlide 4- 22Slide 4-21EXAMPLE4Find thec absolute maximum and minimum values of f(x) = x2/3 on theEXAMPLE3Find the absolute maximum and minimum values of f(x) = 10x(2 In x)interval [2, 3]on the interval [1, e']SolutionSolutionThe first derivativeThe first derivative is)-r3VF() 10(2 Inx) 1010(1 Inx):has no zeros but is undefined at the interior point x = 0, Thbe values of J at this one criti.Theonly critical point in the domain [,']is the poinbhere In x 1,The valuescal point and at the endpoints areof J at this one critical point and at the endpoints areCritical poinr value: J(0) = 0Critical point value:f(e) 10e(2) = (2) = V4Endpoint values:Endpoint values:/(1) = 10(2 In 1) = 20J(3) = (3)p = V9ftel) = 10e (2 2 In c) = 0.Wecanscefrom thislisthatthe functionabolutemaximumvalueisV2.08andiWe can see from this list that the function's absolute maximum value is 10e 27.2; it oc-occurs at the right endpoint x = 3. The absolute minimum value is 0, and it occurs at thecurs at the critical interior point x = e. The absolute minimum value is O and occurs at theimteriorpoint0 where thegraph hasa cusp (Figure4.9).right endpoint x = -Slide 4- 23slide 4- 24
2016/11/15 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 19 2. Method for Finding Absolute Extreme Values on a Closed Interval Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 21 Critical Points Need Not Give Extreme Values Slide 4 - 22 Slide 4 - 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 24
ExercisesFind the absolute maximum and minimum values ofeach function20,2≤153on the given interval.Absolute maximum:28. h(x) = -3r2/3, -1 ≤x≤1Localalso a local maximummaximum230. g(x) = V5 x, V5 ≤x ≤ 037. g(x) = xe, 1 ≤x ≤ 1739. f(o) ↓ + Inx, 0.5 ≤ x 4Absolute minimum;also a local minimumFIGURE 4.9The extreme values ofJ(c) =on[2, 3] ourat x = 0 andx = 3 (Example 3)lide 4- 25Slide 4- 28DEFINITIONSA function J has a local maximum value at an interior point cof its domain if(x) ≤ f(c)for all x in some open interval containing eA function / has a local minimum value at an interior point c of its domain if(x) ≥ f(e)for all x in some open interval containing c3.Local (Relative)ExtremeValuesDEFINITIONSLet y be a funection with domain D.Then f has an absolute (orglobal) maximum value on D at a point c iff(x) ≤ f(c)for all x in Dand an absolute(or global) minimum value on D at ciffor all x in Df(x) ≥ J(c)extremevalueinitsimmediateneighborhood.Slide 4- 27Slide 4- 28Local maximum valueTHEOREM2The First Derivative Theorem for Local Extreme ValuesIfyhasalocalmaximum orminimum value at an interiorpointcofits domainy=fx)and if f' is defined at c, thenI'(c) = 0.TheonlyplaceswhereafunctionfcanpossiblyhaveanSecant slopes 2 0Secant slopes s 0extreme value (local or global) are(never positive)(never negative)1 interior points where f=0 interior points where f"is undefined3 endpoints of the domain offFIGURE4.6A curve with a localmaximum value.The slope at e,simultaneously the limit of nonpositivenumbers and nonnegative mumbers, is zero,Slide 4- 29Slide 4-30
2016/11/15 5 Slide 4 - 25 Slide 4 - 26 Exercises Find the absolute maximum and minimum values of each function on the given interval. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison -Wesley -Wesley Slide 4 - 27 3. Local (Relative) Extreme Values Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 28 Local extrema are also called relative extrema. A function f has a local extrema at an endpoint c if the appropriate inequality holds for all x in some half-open interval containing c. An absolute extremum is also a local extremum, because being an extreme value overall makes it an extreme value in its immediate neighborhood. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 29 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4 - 30 The only places where a function f can possibly have an extreme value (local or global) are interior points where f ' =0 interior points where f ' is undefined endpoints of the domain of f