EXERCISE - MAXIMA-MINIMA CHECK YOUR GRASP. f() 5 () 75 f'() 5. () 75 75.() 7. 5 + 5. () 7 {} 5 () 7 ( ) 5. f() 9a + a +, a > f'() 6 8a + a 6( a + a ) 6( a) ( a) p a, q a a a + + a a a (rjctd) or a a 6. f'() ( +. + 6. +...+. 98 } + 7. f() + 6 + a + b f'() g() + + a f''() 6 + ( + ) f''() > f"() is incrasing f' () at actl on point. Th givn function has actl on trmum.. f() { + cos} sin sin sin f'() cos cos cos cos ( cos )(cos ) + / f 8 8 f() f() minimum valu. Lt th lin b () + ( ) + + maimum valu Sum of intrcpt l + +, > + + 5. A. + + + 5 9 qualit holds for Rquird lin is + O n P + 6 (,/) n A' at ( n) A" A" < at maima A ma 6. a + b ' a + b '' 6a + b for point of inflction '' b a a + b...(i) (as ) point satisf th curv also, so a + b...(ii) from (i) & (ii) 9 a, b. f() ( + ) + f'() 6 6( + ) + D > 6( + ).. > ( ) > so rquird st is option (A,C,D)
EXERCISE -. f() p + (p ) + f'() { p + (p) (p+)} f'() { (p )} { (p +)} < p < and < p + < p (, ). f'() (6 ) ( )( + ) + + / ½ f sin f( + ) f 6 + 9 BRAIN TEASERS f() 9 / ½ f(), Min {f(t) : t } ; f,. Th solution st of th inqualit < < < 5 6 f() + a f'() a 5. f() (a ) (a + ) a + If a > < < a / a/ + If a < < a < a / a/ f() 8. f() t dt t dt u du (Put t u) f() f() 'f' is odd function. Chck othr options., sin sin f'() cos ( sin ) sin ( sin )( sin ) cos sin ( sin ) +sin /. a 9 P C sin sin sin Q s sin + sin sin sin r s sin Maimiz 'r'.. Lt 'a' b th sid of bas. R a a. cos volum bas ara hight. cos sin f() {cos sin} f'() / sin {cos.(sin ) + cos } { tan }cos for ma. volum, tan altitud.sin R cos
EXERCISE - Match th Column :. a + b + c Points A, B and D lis on th curv. a b + c a b + c a + b + c 7 Solving th quations w gt a b c. + + To maimiz ara of ABCD, w maimiz ara (BCD). A (,) B (,) / / To maimiz Ara(BCD) w hav to maimiz h (as shown in figur) for maimum h Slop of BD Slop of tangnt at C 7 ( + ) h C D(,7) + + 7 7 C, On th basis of this th coloumns can b matchd. Assrtion and Rason :. / 7+8 MISCELLANEOUS TYPE QUESTIONS St. II is fals. St. I f() has maima at () / & 7 is th closst natural numbr. a n has gratst valu for n 7. Comprhnsion # : Lt n f() Lt n f()...() for limit to ist Lt f() f() a 6 + a 5 + a Also f'() f'() f'() f'() 6a 5 + 5a + a (6a + 5a + a ) f'() a + a + a...() f'() 6a + 5a + a...() Considr q n. () f() n Lt n f() lim Lt a Putting a in () & () a + a 8 6a + 5a 8 on solving this w gt a 5, a 6 5 a a a From figur st. I is fals, bcaus f(h)< f() st. II is obviousl tru.. St. II :- f() f'() + () / ( ) ( ) () / ( ) ( ) f() 6 5 5 + f'() 5 + 8 ( + ) () () + + 5 On th abov basis th answrs can b givn.
EXERCISE - [A] CONCEPTUAL SUBJECTIVE EXERCISE. ( b ) f() ( ) ( + ) f'() {( ) ( + ) + ( ) ( + )} ( ) ( + ) { + + } ( ) ( + ) (5 + ) + 5. P R r S r Q 5 + cos r r cos 5 ( c ) f() n f'() + n + f"() > concav up Lt n, Lt n A(SRQ) ƒ() (cos).sin cos sin Now maimiz f() a a a b. f'() b b (R R (R + R )) f'() a + b f() a + b + c 6. 9. ½ ½ Givn 8 Printd ara (,) ¾ ¾ f() ( ) ( ) 8 Now maimiz th ara. B P APQ ~ ABC A 6 6 ( ) A ( ) Now maimiz th ara. Q C f() is maimum at 5 5 f ' 5a + b f() c, f() a + b + c a, 5 b, c 5 f(). a + b + a c...(i) d d a + b a d d d (a b) d b a slop of normal b a a b (, ) slop of lin joining origin & point (, ) minimum distanc is along normal. b a so a b or...(ii) from (i) & (ii) rquird points ar for ; for sinc ab< c c, (a b) (a b) & c c, (a b) (a b) c c, (a b) (a b) not possibl
EXERCISE - [B] BRAIN STORMING SUBJECTIVE EXERCISE. f'(). cos sin + + + + A (,) B(,) A(,-) ' To minimiz th primtr. AM + MB is to b minimizd. i.. A'M + MB is to b minimizd. [whr A' is imag of A in.] Obviousl A'M + MB is minimizd. whn A',M and B ar collinar. i.. M coincid with origin. M (, ) M. f() + 5 log (m) / + + f'() ( ) for f() to hav ral & distinct roots f().f() < Solving this w gt th rquird st of m. 9. t t a t t sin sin sin( ) a a t sin sin, t a sin ( + ) sin to maimiz primtr w maimiz t + t t + t f() a {sin() + sin(+)} sin a sin cos sin for maimum sin. f() sin + sin f'() sincos(sin + ) f''() 6sincos sin + cos f'() sin or cos or cos if sin sin < sin < < < sin < < othrwis thr is onl on critical point. If >, thn f''() > point of minima & f'() changs sign from positiv to ngativ for sin (point of maima). If < thn is a point of maima whil sin, {} is a point of minima. Thus for function has actl on maima & actl on minima. 6. Lt th vrtics L, M, N of th squar S b (, ), (, ) & (, ) rspctivl & th vrt O b origin. Lt th co-ordinat of vrtics A, B, C, D of th quadrilatral b (p, )(, q)(r, ) & (, s) Thn a ( p) + q N C M b ( q) + ( r) c ( s) + r D B d p + s O A Thus a + b + c + d ( p) + q + ( q) + ( r) + ( s) + r + p + s Lt f() + ( ) f'() ( ) f'() / f''() f() is minimum at / & ma. valu of f() occur at, / f() So a + b + c + d L
EXERCISE - 5 [A]. f() 9a + a + a > f'() 6 8a + a f"() 8a for maimum or minimum 6 8a + a a + a a or a maimum at a and minimum at a (a > ) givn) p a, q a p q a a a(a ) a. f() + f'() ± f"() minimum at. u a cos b sin + a sin b cos u a + b + (a cos b sin ) (a sin b cos ) u a + b + a cos sin a b cos a b sin b sin cos u a + b + a b ( sin cos ) a cos sin b cos sin a + b + a b (a b a b ) sin cos JEE-[MAIN] : PREVIOUS YEAR QUESTIONS 5. A triangular park (cos)(sin) sin ma. 6. Using A.M. G.M. p q p.q pq B (p + q) p + q + pq (p + q) A cosq sin 8. Graph of P() undr givn conditions. It is clar that P() has ma. at but not minimum at. 9. Point (t, t) is on th parabola Its distanc from t t d(t) d'(t) [t] C a + b + sin a b (a b ) t a + b + a b (a b ) sin u is maimum whn sin u is minimum whn sin u (ma.) u (min.) (a + b ) (a + b) a + b a b ab a + b ab (a b). f() f '() For maimum or minimum, f '() ± Now, f"() at, f"() > and f"() < So, thr ists a local minimum at. d"(t) > d(t) is min at t Its valu d d 8. f has a local minimum at lim f() f() k + k k Dirct ions : Qu st ions numbr 86 to 9 ar Assrtion - Rason tp qustions. Each of ths qustions contains two statmnts :
Statmnt (Assrtion) and Statmnt (Rason). Each of ths qustions also has four altrnativ choics, onl on of which is th corrct answr. You hav to slct th corrct choic.. f'() sin f'() & f'() ar. f '() + local maimum at and local minimum at. At ƒ() and for h and h (h ; h > ) tan > Function has a minima at Statmnt is tru. tan ; Now ƒ() ; sc -tan ; ƒ '() ; ƒ ' () Statmnt- is also tru.. V r Initiall r 5, r r + 5 r r 5m So, (5) 8() + C C 5 r 5t + 75...(ii) At tim t 9 min r 9 m from q. (i) dr 8 /9 dt (9) t9 (Ngativ sign shows dcrmnt in radii) 5. f'() + b + a f'() b + a...() f'() + b + a... () solv () & () a, b st : is tru f''() f"() < f"() < Local maimum at & 6. ƒ() + + k ƒ '() 6 + > ƒ is incrasing function (alwas iv) ƒ() has actl on ral root (as it is an odd dgr polnomial) Now dv dt (r ) dr dt 7 r dr dt dr dt 8 r...(i) r dr 8 dt At t, r 5 m r 8 t + C
EXERCISE - 5 [B] JEE-[ADVANCED] : PREVIOUS YEAR QUESTIONS. ƒ () ( + b ) + b + It is a quadratic prssion with coff. of + b >. ƒ () rprsnts an upward parabola whos min valu is D a b ( b ) m(b) ( b ) For rang of m(b) : b b Thus m(b) (, ], D bing th discriminant. m(b) b > also b + b 7. Lt p() a + b + c + d p() a + b c + d...(i) p() 6 a + b + c + d 6...(ii) p() has maima at p'() a b + c...(iii) p'() has min. at p"() 6a + b...(iv) Solving (i), (ii), (iii) and (iv) w gt From (iv) b a From (iii) a + 6a + c c 9a From (ii) a a 9a + d 6 da 6 From (i) a a + 9a + a 6 6a 6 a b, c 9, d 5 p() 9 + 5 p'() 6 9 ( + ) ( ) is a pt. of ma (givn) and is at pt. of min. [ ma and min occur altrnativl] pt. of local ma is (, ) and pt. of local min is (, ) And distanc btwn thm is [ ( )] ( ) 6 9. (a,b) g() g'() ƒ() g'() at + n ƒ (t) dt & g"() g"( + ln) and g"() g() has local ma. at + ln and local min. at.. (A) + ( ) + is ral ; so D > + 6, so minimum valu (B) (A + B)(A B) (A B)(A + B) AB BA as A is smmtric & B is skw smmtric (AB) t AB k, (C) a log log a log (D) a ( k ) Now ( k log ) k <. < so k is possibl sin cos k cos cos n ± n ± ± n k ( ± ) vn intgr 65
. f() f'() a a a( ) ( a ) a f"() (a ) and f"() a and f"() (a ) (a + ) f"() + ( a) f"(). As whn (, ), f'() < a( a) ( a ) so f() is dcrasing on (, ) at a f"() (a ) at.. g() g'() f '(t) t dt so local minima f '( ) a( ) ( a ) ( ) g'() > whn > g'() < whn < 5. f() 5 + 6 8 St A 9 6. Lt 9 + (5) () [, 5] Now, f'() 6 + 6 5 + 6, and f(), In th st A, f() is incrasing f() ma f(5).5 5.5 + 6.58 7 p p() Lt Lt p() a + b + c Lt p c p() a + b + in (, ) Now, p'() a + b + p'(), p'() a + b + a + b + a, b p() + p() 8 + 7. If [, ] thn Add to all sids h() g() f()... (i) whr, f() f'() ( ) f() has a maima at h() a h '() ( ) h() has a maima at c h() g() f() g() also has a maimum valu at a b c 8. f'()(9)() () ( ) R f() n(g()) R g() f() g'() f().f'() f'() incrasing dcrasing dcrasing incrasing incrasing + 9 + + local maimum at 9, hnc onl point. 9. f() + ( + ) ( ) f() + + + + < < < f has 5 points whr it attains ithr a local maimum or local minimum.
. Lt P'() k( ) ( ) k( + ) P() k c P() 6 k c 6 P()...() c...() b (i) and (ii) k P'() ( ) ( ) P'() 9. Whr P 8 + 5 + 8 + 5 & is constant. ƒ() (a + b) b a a, a b min (a, b) b, a b whr a, b + Local maima and minima at, & 5 8 Lt rmovd lngth from ach sids is Rmovd ara is 5 V (8 ) (5 ) V l 6 + dv 9 d Put 5 6 + + 6 + 5 ( ) (6 5) & 5 6 d v 9 9 d d v at d 5 d v at 6 d (rjctd)