CHAPTER 4 APPLICATIONS OF DERIVATIVES

Transcription

1 CHAPTER APPLICATIONS OF DERIVATIVES. EXTREME VALUES OF FUNCTIONS. A absolute miimum at c, a absolute maimum at b. Theorem guaratees the eistece of such etreme values because h is cotiuous o [aß b].. A absolute miimum at b, a absolute maimum at c. Theorem guaratees the eistece of such etreme values because f is cotiuous o [aß b].. No absolute miimum. A absolute maimum at c. Sice the fuctios domai is a ope iterval, the fuctio does ot satisfy the hypotheses of Theorem ad eed ot have absolute etreme values.. No absolute etrema. The fuctio is either cotiuous or defied o a closed iterval, so it eed ot fulfill the coclusios of Theorem. 5. A absolute miimum at a ad a absolute maimum at c. Note that y g() is ot cotiuous but still has etrema. Whe the hypothesis of Theorem is satisfied the etrema are guarateed, but he the hypothesis is ot satisfied, absolute etrema may or may ot occur. 6. Absolute miimum at c ad a absolute maimum at a. Note that y g() is ot cotiuous but still has absolute etrema. Whe the hypothesis of Theorem is satisfied the etrema are guarateed, but he the hypothesis is ot satisfied, absolute etrema may or may ot occur. 7. Local miimum at aß! b, local maimum at aß! b 8. Miima at aß! b ad aß! b, maimum at a!ß b 9. Maimum at a!ß & b. Note that there is o miimum sice the edpoit aß! b is ecluded from the graph. 0. Local maimum at aß! b, local miimum at aß! b, maimum at aß b, miimum at a!ß b. Graph (c), sice this the oly graph that has positive slope at c.. Graph (b), sice this is the oly graph that represets a differetiable fuctio at a ad b ad has egative slope at c.. Graph (d), sice this is the oly graph represetig a futio that is differetiable at b but ot at a.. Graph (a), sice this is the oly graph that represets a fuctio that is ot differetiable at a or b. 5. f has a absolute mi at 0 but does ot have a absolute ma. Sice the iterval o hich f is defied,, is a ope iterval, e do ot meet the coditios of Theorem.

2 68 Chapter Applicatios of Derivatives 6. f has a absolute ma at 0 but does ot have a absolute mi. Sice the iterval o hich f is defied,, is a ope iterval, e do ot meet the coditios of Theorem. 7. f has a absolute ma at but does ot have a absolute mi. Sice the fuctio is ot cotiuous at, e do ot meet the coditios of Theorem. 8. f has a absolute ma at but does ot have a absolute mi. Sice the fuctio is ot cotiuous at 0, e do ot meet the coditios of Theorem. 9. f has a absolute ma at ad a absolute mi at. Sice the iterval o hich f is defied, 0, is a ope iterval, e do ot meet the coditios of Theorem. 0. f has a absolute ma at 0 ad a absolute mi at ad. Sice f is cotiuous o the closed iterval o hich it is defied, Ÿ Ÿ, e do meet the coditios of Theorem. 9. f() 5 Ê f () Ê o critical poits; f( ), f() Ê the absolute maimum is at ad the absolute miimum is 9 at

3 Sectio. Etreme Values of Fuctios 69. f() Ê f () Ê o critical poits; f( ) 0, f() 5 Ê the absolute maimum is 0 at ad the absolute miimum is 5 at. f() Ê f () Ê a critical poit at 0; f( ) 0, f(0), f() Ê the absolute maimum is at ad the absolute miimum is at 0. f() % Ê f () Ê a critical poit at 0; f( ) 5, f(0), f() Ê the absolute maimum is at 0 ad the absolute miimum is 5 at 5. F() Ê F (), hoever 0 is ot a critical poit sice 0 is ot i the domai; F(0.5), F() 0.5 Ê the absolute maimum is 0.5 at ad the absolute miimum is at F() Ê F (), hoever 0 is ot a critical poit sice 0 is ot i the domai; F( ), F( ) Ê the absolute maimum is at ad the absolute miimum is at 7. h() È Î Î Ê h () Ê a critical poit at 0; h( ), h(0) 0, h(8) Ê the absolute maimum is at 8 ad the absolute miimum is at

4 70 Chapter Applicatios of Derivatives Î Î 8. h() Ê h () Ê a critical poit at 0; h( ), h(0) 0, h() Ê the absolute maimum is 0 at 0 ad the absolute miimum is at ad at 9. g() È a b Î Ê g () Î a b ( ) È Ê critical poits at ad 0, but ot at because is ot i the domai; g( ) 0, g(0), g() È Ê the absolute maimum is at 0 ad the absolute miimum is 0 at 0. g() È5 Î Î a& b a5 b ( ) Ê g () ˆ Ê critical poits at È5 È & ad 0, but ot at È 5 because È 5 is ot i the domai; f Š È5 0, f(0) È5 Ê the absolute maimum is 0 at È5 ad the absolute miimum is È5 at 0 ˆ ˆ ˆ 5 6 ) at ). f( )) si ) Ê f ()) cos ) Ê ) is a critical poit, but ) is ot a critical poit because is ot iterior to the domai; f, f, f Ê the absolute maimum is at miimum is ad the absolute. f( )) ta ) Ê f ()) sec ) Ê f has o critical poits i ˆ ß. The etreme values therefore occur at the edpoits: f ˆ È ad f ˆ Ê the absolute maimum is at ) ad the absolute miimum is È at )

5 Sectio. Etreme Values of Fuctios 7. g() csc Ê g () (csc )(cot ) Ê a critical poit at ; g ˆ, g ˆ, g ˆ Ê the È È È absolute maimum is at ad, ad the absolute miimum is at. g() sec Ê g () (sec )(ta ) Ê a critical poit at 0; g ˆ, g(0), g ˆ Ê the absolute 6 maimum is at ad the absolute miimum is at 0 È È Î 5. f(t) kk t t at b Ê f (t) t t t a b (t) Î Èt Ê a critical poit at t 0; f( ), f(0), f() Ê the absolute maimum is at t 0 ad the absolute miimum is at t kk t È Î 6. f(t) kt 5 k (t 5) a(t 5) b Ê f (t) a (t 5) b ((t 5)) t5 t5 Î È(t 5) kt5k Ê a critical poit at t 5; f(), f(5) 0, f(7) Ê the absolute maimum is at t 7 ad the absolute miimum is 0 at t 5 %Î Î 7. f() Ê f () Ê a critical poit at 0; f( ), f(0) 0, f(8) 6 Ê the absolute maimum is 6 at 8 ad the absolute miimum is 0 at 0 &Î 5 Î 8. f() Ê f () Ê a critical poit at 0; f( ), f(0) 0, f(8) Ê the absolute maimum is at 8 ad the absolute miimum is at Î& Î& 9. g( )) ) Ê g ()) 5 ) Ê a critical poit at ) 0; g( ) 8, g(0) 0, g() Ê the absolute maimum is at ) ad the absolute miimum is 8 at ) Î Î 0. h( )) ) Ê h ()) ) Ê a critical poit at ) 0; h( 7) 7, h(0) 0, h(8) Ê the absolute maimum is 7 at ) 7 ad the absolute miimum is 0 at ) 0. y 6 7 Ê y 6 Ê 6 0 Ê. The critical poit is.. fab 6 Ê f ab Ê 0 Ê a b 0 Ê 0 or. The critical poitss are 0 ad.

6 7 Chapter Applicatios of Derivatives. fab a b Êf ab a b ab a b a b a b a b a b a b a b Ê a b a b 0 Ê or. The critical poits are ad.. gab a b a b Êg ab a b a baba bab a b a ba b a ba b a ba ba b Ê a ba ba b 0 Ê or or. The critical poits are,, ad. 5. y Êy Ê!Ê!Ê ; udefied Ê 0 Ê 0. The domai of the fuctio is a_, 0b a0, _ b, thus 0 is ot i the domai, so the oly critical poit is. a b ab ab ab ab ab 6. fab Êf ab Ê!Ê!Ê 0 or ; udefied Ê a b 0 Ê. The domai of the fuctio is a_, b a, _ b, thus is ot i the domai, so the oly critical poits are 0 ad È Î 6 È È È È Î Î Î 7. y Ê y Ê! Ê 6 0 Ê ; udefied Ê È 0 Ê 0. The critical poits are ad gab È Êg ab Ê 0 Ê 0 Ê ; udefied Ê È 0 È È È 0 Ê 0 or. The critical poits are 0,, ad. 9. Miimum value is at. 50. To fid the eact values, ote that y, hich is zero he È6 9. Local maimum at É Š É ß a0þ86ß 5Þ089 b; local % È * miimum at Š É ß% a0.86ß.9b

7 Sectio. Etreme Values of Fuctios 7 5. To fid the eact values, ote that that y 8 % a ba b, hich is zero he or. Local maimum at aß 7 b; local miimum at ˆ % ß % ( 5. Note that y 5 a 5ba b, hich is zero at 0,, ad 5. Local maimum at a, 08 b; local miimum at a5, 0 b; a0, 0 b is either a maimum or a miimum. 5. Miimum value is 0 he or. È È 5. Note that y, hich is zero at ad is udefied he 0. Local maimum at a0, 0 b; absolute miimum at a, b 55. The actual graph of the fuctio has asymptotes at, so there are o etrema ear these values. (This is a eample of grapher failure.) There is a local miimum at a!ß b.

8 7 Chapter Applicatios of Derivatives 56. Maimum value is at ; miimum value is 0 at ad. 57. Maimum value is at à miimum value is as. 58. Maimum value is at 0à miimum value is as. & % È Î Î 59. y a b a b crit. pt. derivative etremum value % Î &! local ma &! Þ!%! udefied local mi 0

9 Sectio. Etreme Values of Fuctios 75 ) ) È Î Î 60. y a b a % b crit. pt. derivative etremum value! miimum! udefied local ma 0! miimum % 6. y È a b a bè % a% b È% % È% crit. pt. derivative etremum value udefied local ma! È! miimum È! maimum udefied local mi! 6. y È ab È a% bab _5 È È crit. pt. derivative etremum value 0! miimum! %% Î &! local ma && %Þ% udefied miimum! 6. y,, crit. pt. derivative etremum value udefied miimum 6. y,!,! crit. pt. derivative etremum value! udefied local mi! local ma %

10 76 Chapter Applicatios of Derivatives 65. y, 6, crit. pt. derivative etremum value! maimum 5 udefied local mi! maimum 5 &, 66. We begi by determiig hether f is defied at, here f % % Ÿ a b a b ), hä! c Clearly, f a b if, ad lim f a h b. Also, f a b ) if, ad lim f a h b. Sice f is cotiuous at, e have that f a b. Thus, b hä!, f Ÿ a b ), È % a ba) b È%) È a b È È Note that! he, ad )! he. But!Þ)%&, so the critical poits occur at ad Þ&&. crit. pt. derivative etremum value! local ma Þ&&! local mi Þ!(* Î 67. (a) No, sice f ab a b, hich is udefied at. (b) The derivative is defied ad ozero for all Á. Also, f a b! ad fa b! for all Á. (c) No, fa beed ot have a global maimum because its domai is all real umbers. Ay restrictio of f to a closed iterval of the form Òa, b Ó ould have both a maimum value ad miimum value o the iterval. (d) The asers are the same as (a) ad (b) ith replaced by a. *, Ÿ or! Ÿ *, or! 68. Note that fa b. Therefore, f a b. *,! or *,! or (a) No, sice the left- ad right-had derivatives at!, are * ad *, respectively. (b) No, sice the left- ad right-had derivatives at, are ) ad ), respectively. (c) No, sice the left- ad right-had derivatives at, are ) ad ), respectively. (d) The critical poits occur he f a b! (at È ) ad he f a bis udefied (at! ad ). The miimum value is! at, at!, ad at ; local maima occur at Š Èß È ad Š Èß È.

11 È Î Î Sectio. Etreme Values of Fuctios 77 a b Î kk 69. Yes, sice f() kk a b Ê f () a b () is ot defied at 0. Thus it is ot required that f be zero at a local etreme poit sice f may be udefied there. 70. If f(c) is a local maimum value of f, the f() Ÿ f(c) for all i some ope iterval (aßb) cotaiig c. Sice f is eve, f( ) f() Ÿ f(c) f( c) for all i the ope iterval ( bßa) cotaiig c. That is, f assumesa local maimum at the poit c. This is also clear from the graph of f because the graph of a eve fuctio is symmetric about the y-ais. 7. If g(c) is a local miimum value of g, the g() g(c) for all i some ope iterval (aßb) cotaiig c. Sice g is odd, g( ) g() Ÿ g(c) g( c) for all i the ope iterval ( bßa) cotaiig c. That is, g assumes a local maimum at the poit c. This is also clear from the graph of g because the graph of a odd fuctio is symmetric about the origi. 7. If there are o boudary poits or critical poits the fuctio ill have o etreme values i its domai. Such fuctios do ideed eist, for eample f() for. (Ay other liear fuctio f() m b ith m Á 0 ill do as ell.) 7. (a) Vab! & % Vab!!% % a ba! b The oly critical poit i the iterval a!ß & bis at. The maimum value of Va bis at. (b) The largest possible volume of the bo is cubic uits, ad it occurs he uits. 7. (a) f ab a b c is a quadratic, so it ca have 0,, or zeros, hich ould be the critical poits of f. The fuctio fab has to critical poits at ad. The fuctio fab has oe critical poit at!þthe fuctio fab has o critical poits. (b) The fuctio ca have either to local etreme values or o etreme values. (If there is oly oe critical poit, the cubic fuctio has o etreme values.) ds v gt v!! dt! g 0 0 g v! v! v! v! v0 g g 0 g 0 g 0 0 maimum Ÿ Ÿ g! 75. s gt v t s Ê gt v!êt. No sab t s Ítˆ v 0 0 Ít 0 or t. Thus sš gš v Š s s s is the height over the iterval 0 t. di di % 76. dt si t cos t, solvig dt!êta t Êt here is a oegative iteger (i this eercise t is ever egative) Ê the peak curret is È amps. 77. Maimum value is at 5; miimum value is 5 o the iterval Òß Ó; local maimum at a5ß 9b

12 78 Chapter Applicatios of Derivatives 78. Maimum value is o the iterval Ò5ß 7 Ó; miimum value is o the iterval Òß Ó. 79. Maimum value is 5 o the iterval Ò ß _Ñ; miimum value is 5 o the iterval Ð_ß Ó. 80. Miimum value is o the iterval Òß Ó Eample CAS commads: Maple: ith(studet): f := -> ^ - 8*^ + * + ; domai := =-0/5..6/5; plot( f(), domai, color=black, title=sectio. 8(a) ); Df := D(f); plot( Df(), domai, color=black, title=sectio. 8(b) ) StatPt := fsolve( Df()=0, domai ) SigPt := NULL; EdPt := op(rhs(domai)); Pts :=evalf([edpt,statpt,sigpt]); Values := [seq( f(), =Pts )]; Maimum value is.7608 ad occurs at =.56 (right edpoit). Miimum value % is ad occurs at =.8608 (sigular poit). Mathematica: (fuctios may vary) (see sectio.5 re. RealsOly ): <<Miscellaeous `RealOly` Clear[f,] a = ; b = 0/;

13 Sectio. The Mea Value Theorem 79 f[_] = / f[] Plot[{f[], f[]}, {, a, b}] NSolve[f[]==0, ] {f[a], f[0], f[]/.%, f[b]//n I more complicated epressios, NSolve may ot yield results. I this case, a approimate solutio (say. here) is observed from the graph ad the folloig commad is used: FidRoot[f[]==0,{,.}]. THE MEAN VALUE THEOREM f a b f0 a b 0. Whe f() for 0 Ÿ Ÿ, the f ab c Ê c Ê c. Î f a b f0 a b Î 8. Whe f() for 0 Ÿ Ÿ, the f ab c Ê ˆ c Ê c. 0 7 f a bf a Îb Î c. Whe f() for Ÿ Ÿ, the f ab c Ê 0 Ê c.. Whe f() È f f for Ÿ Ÿ, the f ab c Ê Ê c. a b a b È Èc f a bfab È7 ab È7 È7. ad 0.59 are both i the iterval Ÿ Ÿ. 5. Whe f() for Ÿ Ÿ, the f ab c Ê c c Ê c. 6. Whe g() Ÿ Ÿ 0 g a bgab, the g c g c. If 0, the g () g c! Ÿ abê ab Ÿ a b Ê ab Ê c Ê c. Oly c is i the iterval. If! Ÿ, the g () Ê g ab c Ê c Ê c. 7. Does ot; f() is ot differetiable at 0 i ( ß8). 8. Does; f() is cotiuous for every poit of [0ß] ad differetiable for every poit i (0ß). 9. Does; f() is cotiuous for every poit of [0ß] ad differetiable for every poit i (0ß). 0. Does ot; f() is ot cotiuous at 0 because lim f() Á 0 f(0). Ä! c. Does ot; f is ot differetiable at i ( ß0).. Does; f() is cotiuous for every poit of [0ß] ad differetiable for every poit i (0ß).. Sice f() is ot cotiuous o 0 Ÿ Ÿ, Rolles Theorem does ot apply: lim f() lim Á 0 f(). Ä c Ä c. Sice f() must be cotiuous at 0 ad e have lim f() a f(0) Ê a ad Ä! b lim f() lim f() a m b 5 m b. Sice f() must also be differetiable at Ä c Ê Ê Ä b e have lim f () lim f () Ê m Ê m. Therefore, a, m ad b. Ä c k k Ä b = =

14 80 Chapter Applicatios of Derivatives 5. (a) i ii iii iv - (b) Let r ad r be zeros of the polyomial P() a- á a a!, the P(r ) P(r ) 0. Sice polyomials are everyhere cotiuous ad differetiable, by Rolles Theorem P (r) 0 for some r - - betee r ad r, here P () ( ) a á a With f both differetiable ad cotiuous o [aßb] ad f(r ) f(r ) f(r ) 0 here r, r ad r are i [aßb], the by Rolles Theorem there eists a c betee r ad r such that f (c) 0 ad a c betee r ad r such that f (c) 0. Sice f is both differetiable ad cotiuous o [aßb], Rolles Theorem agai applies ad ÐÑ e have a c betee c ad c such that f (c ) 0. To geeralize, if f has zeros i [aßb] ad f is ÐÑ cotiuous o [aß b], the f has at least oe zero betee a ad b. 7. Sice f eists throughout [aß b] the derivative fuctio f is cotiuous there. If f has more tha oe zero i [aß b], say f (r) f (r) 0 for r Á r, the by Rolles Theorem there is a c betee r ad r such that f (c) 0, cotrary to f 0 throughout [aßb]. Therefore f has at most oe zero i [aßb]. The same argumet holds if f 0 throughout [aßb]. r 8. If f() is a cubic polyomial ith four or more zeros, the by Rolles Theorem f () has three or more zeros, f () has o more zeros ad f () has at least oe zero. This is a cotradictio sice f () is a o-zero costat he f() is a cubic polyomial. 9. With f( ) 0 ad f( ) 0 e coclude from the Itermediate Value Theorem that f() has at least oe zero betee ad. The Ê) Ê Ê9 Ê f () 0 for Ê f() is decreasig o [ ß] Êf() 0 has eactly oe solutio i the iterval ( ß ) f() 7 Ê f () 0 o ( _ß 0) Ê f() is icreasig o ( _ß 0). Also, f() 0 if ad f() 0 if 0 Ê f() has eactly oe zero i ( _ß!).. g(t) Èt È t Ê g (t) 0 Ê g(t) is icreasig for t i (!ß_); g() È 0 ad Èt Èt g(5) È5 0 Ê g(t) has eactly oe zero i (!ß_) Þ È t (t) Èt. g(t) t. Ê g (t) 0 Ê g(t) is icreasig for t i ( ß); g( 0.99).5 ad g(0.99) 98. Ê g(t) has eactly oe zero i ( ß).. r( )) ) si ˆ ) 8 Ê r ()) si ˆ ) cos ˆ ) si ˆ ) 0 o ( _ß_ ) Ê r( )) is icreasig o ( _ß _); r(0) 8 ad r(8) si ˆ 8 0 Ê r( ) has eactly oe zero i ( _ß _). ). r( )) ) cos ) È Ê r ()) si ) cos ) si ) 0 o ( _ß_ ) Ê r( )) is icreasig o ( _ß_ ); r( ) cos ( ) È È 0 ad r( ) È 0 Êr( )) has eactly oe zero i ( _ß _). 5. r( )) sec ) 5 Ê r ()) (sec ))(ta )) 0 o ˆ!ß Ê r( )) is icreasig o ˆ!ß ) ; r(0.) 99 ad )% r(.57) 60.5 Ê r( )) has eactly oe zero i ˆ!ß. %

15 Sectio. The Mea Value Theorem 8 6. r( )) ta ) cot ) ) Ê r ()) sec ) csc ) sec ) cot ) 0 o ˆ!ß Ê r( )) is icreasig o ˆ 0 ß ; r ˆ 0 ad r(.57) 5. r( ) has eactly oe zero i ˆ Ê!ß. ) 7. By Corollary, f () 0 for all Ê f() C, here C is a costat. Sice f( ) e have C Ê f() for all. 8. g() 5 Ê g () f () for all. By Corollary, f() g() C for some costat C. The f(0) g(0) C Ê 5 5 C Ê C 0 Ê f() g() 5 for all. 9. g() Ê g () f () for all. By Corollary, f() g() C. (a) f(0) 0 Ê 0 g(0) C 0 C Ê C 0 Ê f() Ê f() (b) f() 0 Ê 0 g() C C Ê C Ê f() Ê f() (c) f( ) Ê g( ) C Ê C Ê C Ê f() Ê f() 0. g() m Ê g () m, a costat. If f () m, the by Corollary, f() g() b m b here b is a costat. Therefore all fuctios hose derivatives are costat ca be graphed as straight lies y m b. %. (a) y C (b) y C (c) y C. (a) y C (b) y C (c) y C. (a) y Ê y C (b) y C (c) y 5 C Î Î. (a) y Ê y C Ê y È C (b) y È C È (c) y C t t 5. (a) y cos t C (b) y si C (c) y cos t si C Î Î Î 6. (a) y ta ) C (b) y ) Ê y ) C (c) y ) ta ) C 7. f() C; 0 f(0) 0 0 C Ê C 0 Ê f() 8. g() C; g( ) ( ) C Ê C Ê g() 9. r( )) 8) cot ) C; 0 r ˆ 8 ˆ cot ˆ C Ê 0 C Ê C Ê r( )) 8) cot ) 0. r(t) sec t t C; 0 r(0) sec (0) 0 C Ê C Ê r(t) sec t t ds dt. v *Þ) t &Ês%Þ* t & t C; at s! ad t! e have C!Ês%Þ* t & t! ds dt. v t Ês t t C; at s % ad t e have C Ês t t ds dt cosatb. v siatb Ês cosatbc; at s! ad t! e have C Ês ds. v cosˆ t Ê s siˆ t t C; at s ad t e have C Ê s siˆ dt

16 8 Chapter Applicatios of Derivatives 5. a Êv t C ; at v! ad t! e have C!Êv t!ês t! t C ; at s & ad t! e have C &Ês t! t & 6. a 9.8 Êv 9.8t C ; at v ad t! e have C Êv*Þ) t Ês%Þ* t t C ; at s! ad t! e have C!Ês%Þ* t t 7. a % sia tb Êv cosa tbc ; at v ad t! e have C!Êv cosa tb Ês sia tbc ; at s ad t! e have C Ês sia tb * 8. a cosˆ t Êv siˆ t C ; at v! ad t! e have C!Êv siˆ t Ês cosˆ t C ; at s ad t! e have C!Êscosˆ t 9. If T(t) is the temperature of the thermometer at time t, the T(0) 9 C ad T() 00 C. From the Mea Value T() T(0) Theorem there eists a 0 t! such that C/sec T (t!), the rate at hich the temperature as chagig at t t as measured by the risig mercury o the thermometer.! 50. Because the truckers average speed as 79.5 mph, by the Mea Value Theorem, the trucker must have bee goig that speed at least oce durig the trip. 5. Because its average speed as approimately kots, ad by the Mea Value Theorem, it must have bee goig that speed at least oce durig the trip. 5. The ruers average speed for the maratho as approimately.909 mph. Therefore, by the Mea Value Theorem, the ruer must have bee goig that speed at least oce durig the maratho. Sice the iitial speed ad fial speed are both 0 mph ad the ruers speed is cotiuous, by the Itermediate Value Theorem, the ruers speed must have bee mph at least tice. d() d(0) 0 d() d(0)!! 0!! 5. Let d(t) represet the distace the automobile traveled i time t. The average speed over 0 Ÿ t Ÿ is. The Mea Value Theorem says that for some 0 t, d (t ). The value d (t ) is the speed of the automobile at time t (hich is read o the speedometer). 5. aab t v ab t Þ Ê vab t Þ t C; at a!ß! be have C! Ê vab t Þ t. Whe t!, the v a! b %) m/sec. b a ab 55. The coclusio of the Mea Value Theorem yields Ê c ˆ a b Ê c Èab. b a c ab b a ab b a 56. The coclusio of the Mea Value Theorem yields c Ê c. 57. f () [cos si ( ) si cos ( )] si ( ) cos ( ) si ( ) si ( ) si ( ) si ( ) 0. Therefore, the fuctio has the costat value f(0) si hich eplais hy the graph is a horizotal lie. & 58. (a) fab a ba ba ba b & % is oe possibility.

17 & % Sectio. The Mea Value Theorem 8 (b) Graphig fab & % ad f ab & & % o Òß Ó by Ò(ß (Ó e see that each -itercept of f a b lies betee a pair of -itercepts of fa b, as epected by Rolles Theorem. (c) Yes, sice si is cotiuous ad differetiable o a_ß_ b. 59. fa b must be zero at least oce betee a ad b by the Itermediate Value Theorem. No suppose that fa b is zero tice betee a ad b. The by the Mea Value Theorem, f a bould have to be zero at least oce betee the to zeros of fa b, but this cat be true sice e are give that f a b Á! o this iterval. Therefore, fa bis zero oce ad oly oce betee a ad b. 60. Cosider the fuctio kab fabga b. ka b is cotiuous ad differetiable o Òa, b Ó, ad sice kab a fab a gab a ad kabb fabbgab b, by the Mea Value Theorem, there must be a poit c i aa, b bhere k ac b!. But sice k ab c f ab c g ab c, this meas that f ab c g ab c, ad c is a poit here the graphs of f ad g have taget lies ith the same slope, so these lies are either parallel or are the same lie. 6. f ab Ÿ for Ÿ Ÿ Ê fa bis differetiable o Ÿ Ÿ Ê f is cotiuous o Ÿ Ÿ Ê f satisfies the f a b f a b f a b f a b coditios of the Mea Value Theorem Ê f ab c for some c i Ê f ab c Ÿ Ê Ÿ Ê f a bf a b Ÿ 6. 0 f a b for all Ê f a beists for all, thus f is differetiable o a, b Ê f is cotiuous o Ò, Ó f a b f a b f a b f a b Ê f satisfies the coditios of the Mea Value Theorem Ê a b f ab c for some c i Ò, Ó Ê 0 Ê 0 fabfab. Sice fabfab Ê fab fab fa b, ad sice 0 fabfab e have fab fa b. Together e have fab fab fa b. 6. Let fab t cos t ad cosider the iterval Ò0, Óhere is a real umber. f is cotiuous o Ò0, Óad f is differetiable o f a b f0 a b a0, b sice f atb si t Êf satisfies the coditios of the Mea Value Theorem Ê f ab c for some c i a0b cos cos cos cos Ò0, Ó Ê si c. Sice Ÿ si c Ÿ Ê Ÿ si c Ÿ Ê Ÿ Ÿ. If 0, Ÿ Ÿ ÊŸ cos Ÿ Êlcos lÿl l. If 0, Ÿ Ÿ Ê cos ÊŸ cos ŸÊ a b Ÿcos ŸÊlcos lÿ l l. Thus, i both cases, e have lcos lÿl l. If 0, the lcos 0 llll0lÿl0 l, thus lcos lÿl lis true for all. 6. Let f() si for a Ÿ Ÿ b. From the Mea Value Theorem there eists a c betee a ad b such that si b si a si b si a si b si a cos c Ê Ÿ Ÿ Ê Ÿ Ê ksi b si ak Ÿ kb a k. ba ba ba 65. Yes. By Corollary e have f() g() c sice f () g (). If the graphs start at the same poit a, the f(a) g(a) Ê c 0 Ê f() g().

18 8 Chapter Applicatios of Derivatives 66. Assume f is differetiable ad lfabfablÿl lfor all values of ad. Sice f is differetiable, f a b eists ad f a b f a b f a b lim usig the alterative formula for the derivative. Let g, hich is cotiuous for all. a b l l Ä By Theorem 0 from Chapter, f a bf a b f a bf a b lf a bf a bl f a b º lim lim lim. Sice º ¹ ¹ Ä Ä Ä ll lf a bf a bl ll lfabfablÿl lfor all ad Ê Ÿ as log as Á. By Theorem 5 from Chapter, f a b lf a bf a bl lim lim f f. a b l l Ÿ Ê Ÿ Ê Ÿ a b Ä Ä Ÿ f(b) f(a) 67. By the Mea Value Theorem e have b a f (c) for some poit c betee a ad b. Sice b a 0 ad f(b) f(a), e have f(b) f(a) 0 Ê f (c) The coditio is that f should be cotiuous over [aß b]. The Mea Value Theorem the guaratees the eistece of a poit c i (aßb) such that f(b) f(a) b a maimum value o [aßb], ad mi f Ÿ f (c) Ÿ ma f, as required. % % % f (c). If f is cotiuous, the it has a miimum ad 69. f () a cos b Ê f () a cos b a cos si b % a cos b ( cos si ) 0 for 0 Ÿ Ÿ0. Ê f () is decreasig he 0 Ÿ Ÿ0. f(0.) Ê mi f ad ma f. No e have Ÿ 0. Ÿ Ê Ÿ f(0.) Ÿ 0. Ê Ÿ f(0.) Ÿ.. a % b f(0.) 0. % % 70. f () a b Ê f () a b a b 0 for 0 Ÿ 0. Ê f () is icreasig he 0 Ÿ Ÿ 0. Ê mi f ad ma f.000. No e have Ÿ Ÿ.000 Ê 0. Ÿ f(0.) Ÿ Ê. Ÿ f(0.) Ÿ.000. f() f() 7. (a) Suppose, the by the Mea Value Theorem 0 Ê f() f(). Suppose, the by the f() f() Mea Value Theorem 0 Ê f() f(). Therefore f() for all sice f(). f() f() f() f() (b) Yes. From part (a), lim 0 ad lim 0. Sice f () eists, these to oe-sided Ä c Ÿ Ä b limits are equal ad have the value f () Ê f () Ÿ 0 ad f () 0 Ê f () 0. f(b) f(a) b a 7. From the Mea Value Theorem e have f (c) here c is betee a ad b. But f (c) pc q 0 q p has oly oe solutio c. (Note: p Á0 sice f is a quadratic fuctio.). MONOTONIC FUNCTIONS AND THE FIRST DERIVATIVE TEST. (a) f () ( ) Ê critical poits at 0 ad (b) f ± ± Ê icreasig o ( _ß!) ad (ß _), decreasig o (!ß )! (c) Local maimum at 0 ad a local miimum at. (a) f () ( )( ) Ê critical poits at ad (b) f ± ± Ê icreasig o ( _ß ) ad (ß _), decreasig o ( ß ) (c) Local maimum at ad a local miimum at. (a) f () ( ) ( ) Ê critical poits at ad (b) f ± ± Ê icreasig o ( ß ) ad (ß _), decreasig o ( _ß )

19 (c) No local maimum ad a local miimum at Sectio. Mootoic Fuctios ad the First Derivative Test 85. (a) f () ( ) ( ) Ê critical poits at ad (b) f ± ± Ê icreasig o ( _ß ) ( ß ) (ß _), ever decreasig (c) No local etrema 5. (a) f () ( )( )( ) Ê critical poits at, ad (b) f ± ± ± Ê icreasig o ( ß ) ad ( ß _), decreasig o ( _ß ) ad (ß ) (c) Local maimum at, local miima at ad 6. (a) f () ( 7)( )( 5) Ê critical poits at 5, ad 7 (b) f ± ± ± Ê icreasig o ( 5ß ) ad (7 ß _), decreasig o ( _ß 5) ad ( ß 7) & ( (c) Local maimum at, local miima at 5 ad 7 a b 7. (a) f () a b Ê critical poits at 0, ad (b) f )( ± ± Ê icreasig o a_ß bad a ß _ b, decreasig o aß 0 b ad a0ß b 0 (c) Local miimum at abab 8. (a) f () abab Ê critical poits at,,, ad (b) f ± )( ± )( Ê icreasig o a_ß b, aß bad a ß _ b, decreasig o aß b ad aßb (c) Local maimum at ad 9. (a) f () Ê critical poits at, ad 0. (b) f ± )( ± Ê icreasig o a_ß bad a ß _ b, decreasig o aß 0 b ad a0ß b 0 (c) Local maimum at, local miimum at 6 È È 6 È 0. (a) f () Ê critical poits at ad 0 (b) f ( ± Ê icreasig o a ß _ b, decreasig o a0ß b 0 (c) Local miimum at Î. (a) f () ( ) Ê critical poits at ad 0 (b) f ± )( Ê icreasig o ( _ß ) ad (0 ß _), decreasig o ( ß 0) 0 (c) Local maimum at, local miimum at 0 Î. (a) f () ( ) Ê critical poits at 0 ad (b) f ( ± Ê icreasig o ( ß _), decreasig o (0ß ) 0 (c) No local maimum ad a local miimum at. (a) f () asi bacos b, 0 Ÿ Ÿ Ê critical poits at,, ad (b) f Ò ± ± ± Ó Ê icreasig o ˆ ß, decreasig o ˆ 0 ß, ˆ ß ad ˆ ß 0

20 86 Chapter Applicatios of Derivatives (c) Local maimum at ad 0, local miimum at ad 5 7. (a) f () asi cos basi cos b, 0 Ÿ Ÿ Ê critical poits at,,, ad (b) f Ò ± ± ± ± Ó Ê icreasig o ˆ ß ad ˆ 5 7 ß, decreasig o ˆ 0 ß, ˆ ß ad ˆ ß 7 5 (c) Local maimum at 0, ad, local miimum at, ad 5. (a) Icreasig o aß0 bad aß b, decreasig o aß b ad a0ßb (b) Absolute maimum at aß b, local maimum at a0ß b ad aß b; Absolute miimum at aß b, local miimum at a0 ß b 6. (a) Icreasig o aß.5 b, a.5ß b, ad aß b, decreasig o a.5ß.5 b ad aßb (b) Absolute maimum at aß b, local maimum at a.5ß b ad aß b; Absolute miimum at a.5ß b, local miimum at aß0 b ad aß0b 7. (a) Icreasig o aß b, a0.5ß b, ad aß b, decreasig o aß0.5b (b) Absolute maimum at aß b, local maimum at aß b ad aß b; No absolute miimum, local miimum at aßbad a0.5ßb 8. (a) Icreasig o aß.5 b, aß b, ad aß b, decreasig o a.5ß b ad aßb (b) No absolute maimum, local maimum at a.5ß b, aß b ad aß b; No absolute miimum, local miimum at aß0 b ad aßb 9. (a) g(t) t t Ê g (t) t Ê a critical poit at t ; g ±, icreasig o Î ˆ _ß, decreasig o ˆ ß _ (b) local maimum value of g ˆ at t, absolute maimum is at t 0. (a) g(t) t 9t 5 Ê g (t) 6t 9 Ê a critical poit at t ; g ±, icreasig o ˆ _ß, Î decreasig o ˆ ß_ (b) local maimum value of g ˆ 7 7 at t, absolute maimum is at t. (a) h() Ê h () ( ) Ê critical poits at 0, Ê h ± ±, icreasig o ˆ 0 ß, decreasig o ( _ß!) ad ˆ ß _! %Î (b) local maimum value of h ˆ at ; local miimum value of h(0) 0 at 0, o absolute etrema 7. (a) h() 8 Ê h () Š È Š È Ê critical poits at È Ê h, icreasig o Š _ß È ad Š Èß _, decreasig o Š Èß È È È (b) a local maimum is h Š È È at È; local miimum is h Š È È at È, o absolute etrema

21 Sectio. Mootoic Fuctios ad the First Derivative Test 87. (a) f( )) ) ) Ê f ()) 6) ) 6 )( )) Ê critical poits at ) 0, Ê f ± ±,! Î icreasig o ˆ 0 ß, decreasig o ( _ß!) ad ˆ ß _ (b) a local maimum is f ˆ at ), a local miimum is f(0) 0 at ) 0, o absolute etrema. (a) f( )) 6 ) ) Ê f ()) 6 ) Š È ) Š È ) Ê critical poits at ) È Ê f ± ±, icreasig o Š Èß È, decreasig o Š _ß È ad Š Èß _ È È (b) a local maimum is f Š È È at ) È, a local miimum is f Š È % È at ) È, o absolute etrema 5. (a) f(r) r 6r Ê f (r) 9r 6 Ê o critical poits Ê f, icreasig o ( _ß _), ever decreasig (b) o local etrema, o absolute etrema 6. (a) h(r) (r 7) Ê h (r) (r 7) Ê a critical poit at r 7 Ê h ±, icreasig o ( ( _ß 7) ( (ß _), ever decreasig (b) o local etrema, o absolute etrema % 7. (a) f() 8 6 Ê f () 6 ( )( ) Ê critical poits at 0 ad Ê f ± ± ±, icreasig o ( ß!) ad (ß _), decreasig o ( _ß ) ad (!ß )! (b) a local maimum is f(0) 6 at 0, local miima are f a b 0 at, o absolute maimum; absolute miimum is 0 at % 8. (a) g() Ê g () ) ( )( ) Ê critical poits at 0,, Ê g ± ± ±, icreasig o (0ß ) ad (ß _), decreasig o ( _ß 0) ad ( ß )! (b) a local maimum is g() at, local miima are g(0) 0 at 0 ad g() 0 at, o absolute maimum; absolute miimum is 0 at 0, % & 9. (a) H(t) t t Ê H (t) 6t 6t 6t ( t)( t) Ê critical poits at t 0, Ê H ± ± ±, icreasig o ( _ß ) ad (0ß ), decreasig o ( ß 0) ad (ß _)! (b) the local maima are H( ) at t ad H() at t, the local miimum is H(0) 0 at t 0, absolute maimum is at t ; o absolute miimum & % 0. (a) K(t) 5t t Ê K (t) 5t 5t 5t ( t)( t) Ê critical poits at t 0, Ê K ± ± ±, icreasig o ( ß 0) (0ß ), decreasig o ( _ß ) ad ( ß _)! (b) a local maimum is K() 6 at t, a local miimum is K( ) 6 at t, o absolute etrema È È È. (a) f() 6È Ê f () Ê critical poits at ad 0 Ê f ( ±, icreasig o a0, _ b, decreasig o a, 0b 0 (b) a local miimum is fa0b 8, a local ad absolute maimum is fab, absolute miimum of 8 at 0

22 88 Chapter Applicatios of Derivatives È È È. (a) g() Ê g () Ê critical poits at ad 0 Ê g Ð ±, icreasig o 0,, decreasig o, _ 0 a b a b (b) a local miimum is fa0b, a local maimum is fab 6, absolute maimum of 6 at. (a) g() È8 Î Î a8 b Ê g () a8 b ˆ Î a) b ( ) Î ( )( ) ÊŠ È Š È Ê critical poits at, È Ê g ( ± ± Ñ, icreasig o ( ß ), decreasig o È È Š È ß ad Š ß È (b) local maima are g() at ad g Š È 0 at È, local miima are g( ) at ad g Š È 0 at È, absolute maimum is at ; absolute miimum is at. (a) g() È Î Î 5 (5 ) Ê g () (5 ) ˆ Î (5 ) ( ) 5( ) Ê critical poits at È5 0, ad 5 Ê g ± ± Ñ, icreasig o (0ß ), decreasig o ( _ß!) ad (%ß &)! % & (b) a local maimum is g() 6 at, a local miimum is 0 at 0 ad 5, o absolute maimum; absolute miimum is 0 at 0, 5 ( ) a b() ( )( ) () ( ) 5. (a) f() Ê f () Ê critical poits at, Ê f ± )( ±, icreasig o ( _ß ) ad ( ß _), decreasig o (ß ) ad (ß ), discotiuous at (b) a local maimum is f() at, a local miimum is f() 6 at, o absolute etrema a b (6) a b a b a b 6. (a) f() Ê f () Ê a critical poit at 0 Ê f ±, icreasig o ( _ß!) (!ß _), ad ever decreasig! (b) o local etrema, o absolute etrema Î %Î Î Î 8 Î ( ) 7. (a) f() ( 8) 8 Ê f () Ê critical poits at 0, Ê f ± )(, icreasig o ( ß!) (!ß _), decreasig o ( _ß )! (b) o local maimum, a local miimum is f( ) 6 È 7.56 at, o absolute maimum; absolute miimum is 6 È at Î Î &Î Î 5 Î 0 Î 5( ) È 8. (a) g() ( 5) 5 Ê g () Ê critical poits at ad 0 Ê g ± )(, icreasig o ( _ß ) ad (!ß _), decreasig o ( ß!)! (b) local maimum is g( ) È.76 at, a local miimum is g(0) 0 at 0, o absolute etrema Š È7 Š È7 Î (Î Î 7 %Î Î 9. (a) h() a b Ê h () Ê critical poits at 0, È Ê h ± )( ±, icreasig o Š _ß ad ß _, decreasig o 7 È Š 7 È 7 Î È(! Î È( È7 È7 Š ß! ad Š!ß È

23 Sectio. Mootoic Fuctios ad the First Derivative Test 89 È È È7 7(Î È7 È7 7(Î (b) local maimum is h Š. at, the local miimum is h Š., o absolute etrema Î )Î Î &Î Î 8( )( ) È (a) k() a b Ê k () Ê critical poits at 0, Ê k ± )( ±, icreasig o ( ß!) ad (ß _), decreasig o ( _ß )! ad (!ß ) (b) local maimum is k(0) 0 at 0, local miima are k a b at, o absolute maimum; absolute miimum is at. (a) f() Ê f () Ê a critical poit at Ê f ± ] ad fab ad fab 0 a local maimum is at, a local miimum is 0 at. (b) There is a absolute maimum of at ; o absolute miimum. (c). (a) f() ( ) Ê f () ( ) Ê a critical poit at Ê f ± ] ad f( ) 0, f(0)! Ê a local maimum is at 0, a local miimum is 0 at (b) o absolute maimum; absolute miimum is 0 at (c). (a) g() Ê g () ( ) Ê a critical poit at Ê g [ ± ad g(), g() 0 Ê a local maimum is at, a local miimum is g() 0 at (b) o absolute maimum; absolute miimum is 0 at (c). (a) g() 6 9 Êg () 6 ( ) Ê a critical poit at Ê g [ ± ad % g( ), g( ) 0 Ê a local maimum is 0 at, a local miimum is at

24 90 Chapter Applicatios of Derivatives (b) absolute maimum is 0 at ; o absolute miimum (c) 5. (a) f(t) t t Ê f (t) t ( t)( t) Ê critical poits at t Ê f [ ± ± ad f( ) 9, f( ) 6, f() 6 Ê local maima are 9 at t ad 6 at t, a local miimum is 6 at t (b) absolute maimum is 6 at t ; o absolute miimum (c) 6. (a) f(t) t t Ê f (t) t 6t t(t ) Ê critical poits at t 0 ad t Ê f ± ± ] ad f(0) 0, f(), f() 0 Ê a local maimum is 0 at t 0 ad t, a! local miimum is at t (b) absolute maimum is 0 at t 0, ; o absolute miimum (c) 7. (a) h() Ê h () ( ) Ê a critical poit at Ê h [ ± ad! h(0) 0 Ê o local maimum, a local miimum is 0 at 0 (b) o absolute maimum; absolute miimum is 0 at 0 (c)

25 Sectio. Mootoic Fuctios ad the First Derivative Test 9 8. (a) k() Ê k () 6 ( ) Ê a critical poit at Ê k ± ] ad k( ) 0, k(0) Ê a local maimum is at 0, o local miimum! (b) absolute maimum is at 0; o absolute miimum (c) 9. (a) fab È5 Ê f ab Ê critical poits at 0, 5, ad 5 È5 Ê f Ð ± Ñ, fa5b 0, fa0b 5, fa5b 0 Ê local maimum is 5 at 0; local miimum of 0 at ad 5 (b) absolute maimum is 5 at 0; absolute miimum of 0 at 5 ad 5 (c) 50. (a) fab È, Ÿ _Êf ab Êoly critical poit i Ÿ _ is at È Ê f [, fab 0 Ê local miimum of 0 at, o local maimum (b) absolute miimum of 0 at, o absolute maimum (c) 5. (a) ga b, 0 Ÿ Ê g ab Ê oly critical poit i 0 Ÿ is È 0.68 a b È Ê g [ ), g.866 local miimum of at, local l Š È Ê È È6 È6 maimum at 0. È È 6 (b) absolute miimum of at È, o absolute maimum È

26 9 Chapter Applicatios of Derivatives (c) 8 a b 5. (a) ga b, Ÿ Ê g ab Ê oly critical poit i Ÿ is 0 Ê g ( l ], ga0b 0 Ê local miimum of 0 at 0, local maimum of at. 0 (b) absolute miimum of 0 at 0, o absolute maimum (c) 5. (a) fab si, 0 Ÿ Ÿ Ê f ab cos, f () 0 Ê cos 0 Ê critical poits are ad Ê f [ ± ± ], fa0b 0, fˆ, fˆ, f 0 Ê local maima are at ad 0 0 ab at, ad local miima are at ad 0 at 0. (b) The graph of f rises he f 0, falls he f 0, ad has local etreme values here f 0. The fuctio f has a local miimum value at 0 ad, here the values of f chage from egative to positive. The fuctio f has a local maimum value at ad egative., here the values of f chage from positive to 5. (a) fab si cos, 0 Ÿ Ÿ Ê f ab cos si, f () 0 Ê ta Ê critical poits are ad 7 Ê f [ ], f 0, f, f, f local maima are 0 ± ± 7 a b ˆ È ˆ È a b Ê 7 È 7 È at ad at, ad local miima are at ad at 0. (b) The graph of f rises he f 0, falls he f 0, ad has local etreme values here f 0. The fuctio f has a local miimum value at 0 ad 7, here the values of f chage from egative to positive. The fuctio f has a local maimum value at ad egative., here the values of f chage from positive to

27 Sectio. Mootoic Fuctios ad the First Derivative Test (a) fab È cos si, 0 f ab È Ÿ Ÿ Ê si cos, f () 0 Ê ta Ê critical poits are ad 6 Ê f [ ± ± ], fa0b È, fˆ, fˆ, f Ê local a b È 6 6 È maima are at ad at, ad local miima are at ad È at 0. (b) The graph of f rises he f 0, falls he f 0, ad has local etreme values here f 0. The fuctio f has a local miimum value at 0 ad 7 6, here the values of f chage from egative to positive. The fuctio f has a local maimum value at ad 6 egative., here the values of f chage from positive to 56. (a) fab ta, Ê f ab sec, f () 0 Ê sec Ê critical poits are ad Ê f ( ± ± ), fˆ, fˆ Ê local maimum is at, ad local miimum is at. (b) The graph of f rises he f 0, falls he f 0, ad has local etreme values here f 0. The fuctio f has a local miimum value at, here the values of f chage from egative to positive. The fuctio f has a local maimum value at, here the values of f chage from positive to egative. È 57. (a) f() si ˆ Ê f () cos ˆ, f () 0 Ê cos ˆ Ê a critical poit at Ê f [ ± ] ad f(0) 0, f ˆ È, f( ) Ê local maima are 0 at 0 ad! Î at, a local miimum is È at (b) The graph of f rises he f 0, falls he f 0, ad has a local miimum value at the poit here f chages from egative to positive. 58. (a) f() cos cos Ê f () si cos si (si )( cos ) Ê critical poits at, 0, Ê f [ ± ] ad f( ), f(0), f( ) Ê a local maimum is at, a local! miimum is at 0

28 9 Chapter Applicatios of Derivatives (b) The graph of f rises he f 0, falls he f 0, ad has local etreme values here f 0. The fuctio f has a local miimum value at 0, here the values of f chage from egative to positive. 59. (a) f() csc cot Ê f () (csc )( csc )(cot ) acsc b acsc b(cot ) Ê a critical poit at Ê f ( ± ) ad f ˆ 0 Ê o local maimum, a local miimum is 0 at! Î% (b) The graph of f rises he f 0, falls he f 0, ad has a local miimum value at the poit here f 0 ad the values of f chage from egative to positive. The graph of f steepes as f () Ä _. 60. (a) f() sec ta Ê f () (sec )(sec )(ta ) sec a sec b(ta ) Ê a critical poit at Ê f ( ± ) ad f ˆ 0 Ê o local maimum, a local miimum is 0 at Î Î% Î (b) The graph of f rises he f 0, falls he f 0, ad has a local miimum value here f 0 ad the values of f chage from egative to positive. 6. h( )) cos ˆ ) Ê h ()) si ˆ ) Ê h [ ], (!ß ) ad ( ß ) Ê a local maimum is at ) 0,! a local miimum is at ) 6. h( )) 5 si ˆ ) 5 Ê h ()) cos ˆ ) Ê h [ ], (!ß 0) ad ( ß 5) Ê a local maimum is 5 at ), a local! miimum is 0 at ) 0 6. (a) (b) (c) (d)

29 6. (a) (b) Sectio. Mootoic Fuctios ad the First Derivative Test 95 (c) (d) 65. (a) (b) 66. (a) (b) 67. The fuctio fab si ˆ has a ifiite umber of local maima ad miima. The fuctio si has the folloig properties: a) it is cotiuous o a_, _ b; b) it is periodic; ad c) its rage is Ò, Ó. Also, for a 0, the fuctio has a a a rage of Ð_, aó Ò a, _Ñ o,. I particular, if a, the Ÿ or he is i Ò, Ó. This meas si ˆ takes o the values of ad ifiitely may times i times o the iterval Ò, Ó, hich occur he,, 5,....,,,.... Thus si Ê ˆ 5 has ifiitely may local maima ad miima i the iterval Ò, Ó. O the iterval Ò0, Ó, Ÿ siˆ Ÿ ad sice 0 e have Ÿ siˆ Ÿ. O the iterval Ò, 0 Ó, Ÿ siˆ Ÿ ad sice 0 e have siˆ. Thus fa b is bouded by the lies y ad y. Sice si ˆ oscillates betee ad ifiitely may times o Ò, Óthe f ill oscillate betee y ad y ifiitely may times. Thus f has ifiitely may local maima ad miima. We ca see from the graph (ad verify later i Chapter 7) that lim si ˆ ad lim si ˆ. The graph of f does ot have ay absolute Ä_ maima., but it does have to absolute miima. Ä_

30 96 Chapter Applicatios of Derivatives 68. f() a b c a ˆ b b b b c a c a ˆ b Š b ac, a parabola hose a a a a a a b ˆ b ˆ b a a a b b b a a a verte is at. Thus he a 0, f is icreasig o ß _ ad decreasig o _ß ; he a 0, f is icreasig o ˆ _ß ad decreasig o ˆ ß _. Also ote that f () a b a ˆ Ê for a 0, f ; for a 0, f ±. bîa bîa 69. f() a b Ê f ab a b, fab Ê a b, f ab 0 Ê a b 0 Ê a, b Ê f() 70. f() a b c d Êf ab a b c, fa0b 0 Êd 0, fab Ê a b c d, f a0b 0 Ê c 0, f ab 0 Ê a b c 0 Ê a, b, c 0, d 0 Ê f(). CONCAVITY AND CURVE SKETCHING. y Ê y ( )( ) Ê y ˆ. The graph is risig o ( _ß ) ad (ß _), fallig o ( ß ), cocave up o ˆ ß _ ad cocave do o ˆ _ß. Cosequetly, a local maimum is at, a local miimum is at, ad ˆ ß is a poit of iflectio. %. y Ê y a b ( )( ) Ê y Š È Š È. The graph is risig o ( ß 0) ad (ß _), fallig o ( _ß ) ad (!ß ), cocave up o Š _ß ad Š ß _ ad È È È È cocave do o Š ß. Cosequetly, a local maimum is at 0, local miima are 0 at, ad 6 6 È 9 È 9 Š ß ad Š ß are poits of iflectio. Î Î Î. y a b Ê y ˆ ˆ a b () a b, y ) ( ± )(! Ê the graph is risig o ( ß! ) ad (ß_), fallig o ( _ß ) ad (!ß) Ê a local maimum is at 0, local Î %Î miima are 0 at ; y a b () ˆ a b (), Éa b % y ± ) ( )( ± Ê the graph is cocave up o Š _ß È ad Š È ß _, cocave È È do o Š ÈßÈ Ê poits of iflectio at Š Èß % È % 9 Î Î 9 Î Î. y a 7 b Ê y a 7b () a b, y ± )( ±! 7 Ê the graph is risig o ( _ß ) ad (ß_), fallig o ( ß ) Ê a local maimum is at, a local 7 7 &Î Î Î &Î &Î miimum is at ; y a b a b, y )( Ê the graph is cocave up o (!ß _), cocave do o ( _ß!) Ê a poit of iflectio at (!ß!).! 5. y si Ê y cos, y [ ± ± ] Ê the graph is risig o ˆ ß, fallig Î Î Î Î È È È È o ˆ ad ˆ ß ß Ê local maima are at ad at, local miima are at ad at ; y si, y [ ± ± ± ] Ê the Î Î! Î Î ad ˆ ß, cocave do o ˆ ß ad ˆ!ß graph is cocave up o ˆ ß! Ê poits of iflectio at ˆ ß, (!ß!), ad ˆ ß 7

31 Sectio. Cocavity ad Curve Sketchig y ta Ê y sec, y ( ± ± ) Ê the graph is risig o ˆ ß ad Î Î Î Î ˆ, fallig o ˆ a local maimum is È at, a local miimum is È ß ß Ê at ; y (sec )(sec )(ta ) asec b(ta ), y ( ± ) Ê the graph is cocave up o ˆ 0 ß, Î! Î cocave do o ˆ ß0 Ê a poit of iflectio at (0ß0) 7. If 0, si kk si ad if 0, si kk si ( ) si. From the sketch the graph is risig o ˆ, ˆ ad ˆ ß!ß ß, fallig o ˆ ß, ˆ ad ˆ ß! ß ; local miima are at ad 0 at! ; local maima are at ad 0 at ; cocave up o ( ß) ad ( ß ), ad cocave do o ( ß 0) ad (!ß ) Ê poits of iflectio are ( ß! ) ad ( ß! ) 8. y cos È Ê y si È, y [ ± ± ± ] Ê risig o Î% Î% & Î% Î È ˆ ad ˆ 5, fallig o ˆ ad ˆ 5 ß ß ß ß Ê local maima are È at, È È È 5È at ad at, ad local miima are È at ad È 5 at ; y cos, y [ ± ± ] Ê cocave up o ˆ ß ad ˆ ß, cocave do o Î Î Î È È ˆ ß Ê poits of iflectio at Š ß ad Š ß 9. Whe y, the y ( ) ad y. The curve rises o (ß_) ad falls o ( _ß ). At there is a miimum. Sice y 0, the curve is cocave up for all. 0. Whe y, the y ( ) ad y. The curve rises o ( _ß) ad falls o ( ß_ ). At there is a maimum. Sice y 0, the curve is cocave do for all.

32 98 Chapter Applicatios of Derivatives. Whe y, the y ( )( ) ad y 6. The curve rises o ( _ß ) (ß _) ad falls o ( ß). At there is a local maimum ad at a local miimum. The curve is cocave do o ( _ß 0) ad cocave up o (!ß _). There is a poit of iflectio at 0.. Whe y (6 ), the y (6 ) () ( )( ) ad y ( ) ( ) ( ). The curve rises o ( _ß ) ( ß_ ) ad falls o (ß ). The curve is cocave do o ( _ß ) ad cocave up o (ß _). At there is a poit of iflectio.. Whe y 6, the y 6 6( ) ad y ( ). The curve rises o (!ß ) ad falls o ( _ß 0) ad (ß _). At 0 there is a local miimum ad at a local maimum. The curve is cocave up o ( _ß ) ad cocave do o (ß _). At there is a poit of iflectio.. Whe y 9 6, the y 9 ( )( B) ad y 66( ). The curve rises o ( ß ) ad falls o ( _ß ) ad ( ß _). At there is a local maimum ad at a local miimum. The curve is cocave up o ( _ß ) ad cocave do o ( ß _). At there is a poit of iflectio. 5. Whe y ( ), the y ( ) ad y 6( ). The curve ever falls ad there are o local etrema. The curve is cocave do o ( _ß ) ad cocave up o (ß _). At there is a poit of iflectio.

33 Sectio. Cocavity ad Curve Sketchig Whe y ( ), the y ( ) ad y 6( ). The curve ever rises ad there are o local etrema. The curve is cocave up o ( _ß ) ad cocave do o ( ß _). At there is a poit of iflectio. % 7. Whe y, the y ( )( ) ad y Š È Š È. The curve rises o ( ß!) ad (ß _) ad falls o ( _ß ) ad (!ß ). At there are local miima ad at 0 a local maimum. The curve is cocave up o Š _ß È Š ß_ ad cocave do o Š ß. At there are poits of iflectio. ad È È È È % 8. Whe y 6, the y Š È Š È ad y ( )( ). The curve rises o Š _ßÈ ad Š!ß È, ad falls o Š È ß! ad Š È ß _. At Èthere are local maima ad at 0 a local miimum. The curve is cocave up o ( ß ) ad cocave do o ( _ß ) ad (ß _). At there are poits of iflectio. % 9. Whe y, the y ( ) ad y ( ). The curve rises o a_ß b ad falls o aß _ b. At there is a local maimum, but there is o local miimum. The graph is cocave up o a!ß b ad cocave do o a_ß! b ad aß _ b. There are iflectio poits at 0 ad. % 0. Whe y, the y 6 ( ) ad y ( ). The curve rises o ˆ ad falls o ˆ ß ß. There is a local miimum at, but o local maimum. The curve is cocave up o ( _ß ) ad (!ß _), ad cocave do o ( ß0). At ad 0 there are poits of iflectio.

34 00 Chapter Applicatios of Derivatives & % %. Whe y 5, the y ( ) ad y ( ). The curve rises o ( _ß! ) ad (%ß_), ad falls o (!ß%). There is a local maimum at 0, ad a local miimum at. The curve is cocave do o ( _ß ) ad cocave up o ( ß_ ). At there is a poit of iflectio. % % , ad y ( ). The curve is risig. Whe y ˆ 5, the y ˆ 5 () ˆ 5 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ o ( _ß ) ad (0 ß_ ), ad fallig o (ß0). There is a local maimum at ad a local miimum at 0. The curve is cocave do o ( _ß %) ad cocave up o (%ß _). At there is a poit of iflectio.. Whe y si, the y cos ad y si. The curve rises o (!ß ). At 0 there is a local ad absolute miimum ad at there is a local ad absolute maimum. The curve is cocave do o (!ß ) ad cocave up o ( ß ). At there is a poit of iflectio.. Whe y si, the y cos ad y si. The curve rises o (!ß ). At 0 there is a local ad absolute miimum ad at there is a local ad absolute maimum. The curve is cocave up o (!ß ) ad cocave do o ( ß ). At there is a poit of iflectio. 5. Whe y È cos, the y È si ad y cos. The curve is icreasig o ˆ!ß ad ˆ 5, ad decreasig o ˆ 5 ß ß. At 0 there is a local ad absolute miimum, at there is a local maimum, at 5 there is a local miimum, ad ad at there is a local ad absolute maimum. The curve is cocave up o ˆ ad ˆ!ß ß, ad is cocave do o ˆ ß. At ad there are poits of iflectio.

35 Sectio. Cocavity ad Curve Sketchig 0 6. Whe y ta, the y sec ad y sec ta. The curve is icreasig o ˆ 6ß 6, ad decreasig o ˆ ad ˆ ß ß 6 6. At 6 there is a local miimum, at 6 there is a local maimum,there are o absolute maima or absolute miima. The curve is cocave up o ˆ 0 ß, ad is cocave do oˆ 0 ß. At 0 there is a poit of iflectio. 7. Whe y si cos, the y si cos cos ad y si. The curve is icreasig o ˆ 0 ß ad ˆ, ad decreasig o ˆ ß ß. At 0 there is a local miimum, at there is a local ad absolute maimum, at there is a local ad absolute miimum, ad at there is a local maimum. The curve is cocave do o ˆ 0 ß, ad is cocave up o ˆ ß. At there is a poit of iflectio. 8. Whe y cos È si, the y si Ècos ad y cos È si. The curve is icreasig o ˆ 0 ad ˆ ß ß, ad decreasig o ˆ ß. At 0 there is a local miimum, at there is a local ad absolute maimum, at there is a local ad absolute miimum, ad at there is a local maimum. The curve is cocave do o ˆ 5 0 ad ˆ ß ß 6, ad is cocave up o ˆ 5 5 ß. At ad there are poits of iflectio Î& %Î& *Î& Whe y, the y ad y. The curve rises o ( _ß _) ad there are o etrema. The curve is cocave up o ( _ß!) ad cocave do o (!ß _). At 0 there is a poit of iflectio. Î& Î& 6 )Î& 0. Whe y, the y 5 ad y 5. The curve is risig o (0 ß_ ) ad fallig o ( _ß! ). At 0 there is a local ad absolute miimum. There is o local or absolute maimum. The curve is cocave do o ( _ß!) ad (!ß _). There are o poits of iflectio, but a cusp eists at 0.

36 0 Chapter Applicatios of Derivatives È a Î a 5 b. Whe y, the y ad y Î. The curve is icreasig o a_ß _ b. There are o local or absolute etrema. The curve is cocave up o a_ß! b ad cocave do o a!ß _ b. At 0 there is a poit of iflectio. È a b a b È 7 a b a b ß ß. Whe y, the y ad y. The curve is decreasig o ˆ ad ˆ. There are o absolute etrrema, there is a local maimum at ad a local miimum at. The curve is cocave up o aß0.9 bad ˆ 0.69, ad cocave do o ˆ ß 0.9 ß ad a0.69ß b. At 0.9 ad 0.69 there are poits of iflectio. Î Î. Whe y, the y ad %Î y. The curve is risig o ( _ß!) ad (ß _), ad fallig o (!ß ). There is a local maimum at 0 ad a local miimum at. The curve is cocave up o ( _ß!) ad (!ß _). There are o poits of iflectio, but a cusp eists at 0. Î& Î&. Whe y 5, the y ˆ Î& 6 )Î& ad y 5. The curve is risig o (0ß) ad fallig o ( _ß 0) ad (ß _). There is a local miimum at 0 ad a local maimum at. The curve is cocave do o ( _ß!) ad (!ß _). There are o poits of iflectio, but a cusp eists at 0. Î 5. Whe y ˆ 5 5 Î &Î, the 5 Î 5 Î 5 Î y ( ) ad 5 %Î 0 Î 5 %Î y ( ). The curve is risig o (!ß ) ad fallig o ( _ß!) ad (ß _). There is a local miimum at 0 ad a local maimum at. The curve is cocave up o ˆ _ß ad cocave do o ˆ ß! ad (0 ß_ ). There is a poit of iflectio at ad a cusp at 0.

37 Sectio. Cocavity ad Curve Sketchig 0 Î &Î Î 6. Whe y ( 5) 5, the 5 Î 0 Î 5 Î y ( ) ad 0 Î 0 %Î 0 %Î y ( ). The curve is risig o ( _ß! ) ad (ß_), ad fallig o (!ß). There is a local miimum at ad a local maimum at 0. The curve is cocave up o ( ß 0) ad (!ß _), ad cocave do o ( _ß ). There is a poit of iflectio at ad a cusp at Whe y È8 a8 b Î, the Î Î Î ( )( ) ÊŠ Š È y a8 b () ˆ a8 b ( ) a8 b a8 b ad y ˆ a8 b ( ) a8 ba8 b ( ) a b Éa8 b. The curve is risig o ( ß ), ad fallig o Š È ß ad Š ßÈ. There are local miima ad È, ad local maima at È ad. The curve is cocave up o Š È ß! ad cocave do o Š!ß È. There is a poit of iflectio at 0. Î 8. Whe y a b, the y ˆ a b ( ) È ÊŠ È Š È ad Î y ( ) a b ( ) ˆ a b ( ) 6( )( ) ÊŠ È Š È. The curve is risig o Î Î Š È ß! ad fallig o Š!ßÈ. There is a local maimum at 0, ad local miima at È. The curve is cocave do o ( ß ) ad cocave up o Š È ß ad Š ßÈ. There are poits of iflectio at. È 6 6 a6 Î b 9. Whe y 6, the y ad È y. The curve is risig o aß0 bad fallig o a0ß b. There is a local ad absolute maimum at 0 ad local ad absolute miima at ad. The curve is cocave do o aß b. There are o poits of iflectio.

38 0 Chapter Applicatios of Derivatives 0. Whe y, the y ad y. The curve is fallig o a_ß0b ad a0ß b, ad risig o a ß_ b. There is a local miimum at. There are o absolute maima or absolute miima. The curve is cocave up o Š È _ß ad a 0, _ b, ad cocave do o Š È, 0. There is a poit of iflectio at È. ( ) a b( ) () ( )( ) ( ) ad ( )( ) a b ( ) ( ) % ( ). Whe y, the y y. The curve is risig o ( _ß ) ad ( ß _), ad fallig o (ß ) ad (ß ). There is a local maimum at ad a local miimum at. The curve is cocave do o ( _ß ) ad cocave up o (ß _). There are o poits of iflectio because is ot i the domai. È Î a b a 5 b. Whe y, the y ad y Î. The curve is risg o a_ß b, a, 0 b, ad a0 ß_ b. There is are o local or absolute etrema. The curve is cocave up o a_ß bad a0, _ b, ad cocave do o a, 0 b. There are poits of iflectio at ad ˆ a b 6ˆ. Whe y, the y ad y. The curve is fallg o _ß a b a b ad a ß_ b, ad is risig o aß b. There is a local ad absolute miimum at, ad a local ad absolute maimum at. The curve is cocave do o Š _ß È ad Š 0, È, ad cocave up o Š È, 0 ad Š È, _. There are poits of iflectio at È, 0, ad È. y 0 is a horizotal asymptote a 5b 00 ˆ a 5b. Whe y, the y ad y. The curve is risg o a_ß0 b, ad is fallig o a0 ß_ b. There is a local ad absolute maimum at 0, ad there is o local or aboslute miimum. The curve is cocave up o Š È ad Š È,, ad cocave do o Š È, 0 ad Š 0, È _ß _. There are poits of iflectio at È È ad. There is a horizotal asymptote of y 0.

39 Sectio. Cocavity ad Curve Sketchig 05, kk 5. Whe y k k, the, kk, kk, kk y ad y. The, k k, kk curve rises o ( ß!) ad (ß _) ad falls o ( _ß ) ad (0ß). There is a local maimum at 0 ad local miima at. The curve is cocave up o ( _ß) ad (ß _), ad cocave do o ( ß ). There are o poits of iflectio because y is ot differetiable at (so there is o taget lie at those poits). Ú, 0 6. Whe y k k Û, 0 Ÿ Ÿ, the Ü, Ú, 0 Ú, 0 y Û, 0, ad y Û, 0. Ü, Ü, The curve is risig o (!ß ) ad (ß _), ad fallig o ( _ß!) ad (ß ). There is a local maimum at ad local miima at 0 ad. The curve is cocave up o ( _ß! ) ad (ß_), ad cocave do o (!ß). There are o poits of iflectio because y is ot differetiable at 0 ad (so there is o taget at those poits). È, 0 7. Whe y Èk k, the È, 0 Ú cî, 0, 0 È y Û ad y., 0 cî ( ) Ü È, 0 Sice lim y ad lim y there is a Ä! c Ä! b cusp at 0. There is a local miimum at 0, but o local maimum. The curve is cocave do o ( _ß! ) ad (!ß _). There are o poits of iflectio. È, 8. Whe y Èk k, the È, Ú cî, ( ) È, y Û ad y., cî ( ) Ü È, Sice lim y ad lim y there is a cusp Ä c Ä b at. There is a local miimum at, but o local maimum. The curve is cocave do o ( _ß %) ad (%ß _). There are o poits of iflectio. 9. y ( )( ), y ± ± Ê risig o ( ß ), fallig o ( _ß ) ad (ß _) Ê there is a local maimum at ad a local miimum at ; y, y ± Î Ê cocave up o ˆ _ß, cocave do o ˆ ß_ Ê a poit of iflectio at