Functions nd Grps. () () (c) - - - O - - - O - - - O - - - - (d) () (f) - - O - 7 6 - - O - -7-6 - - - - - O. () () (c) (d) - - - O - O - O - - O - -. () G() f() + f( ), G(-) f( ) + f(), G() G( ) nd G() is vn. H() f() f( ), H( ) f( ) f(), H( ) H() nd H() is odd. f () + f ( ) f () f ( ) () f () +, wic is t sum of n vn nd n odd function.. () sin ( + π) (-sin ) sin. sin is priodic nd t smllst priod is π. () sin ( ) is not priodic. (c) (i) π (ii) π/
. - - - - O - - - 6. () f( + ) f() g() + g() f() ; g( + ) g() g() f() f() Put, f() f()g(). () ; g() [g()] [f()].() From (), f() [ g()] f() or g() / If g() /, from (), / / [f()], [f()] /, wic s no solution. f(), tn from (), g() [g()] g() or g() Howvr, if f() nd g(), w v g() g( + ) g() g() + f() f(), nc g(), contrdicts wit ssumption. f() nd g() () f (), g () f ( + ) f () f () g( + ) g() g() g() lim lim, lim lim, lim lim f ( + ) f () f ()g() + g()f () g() f () (i) f '() lim lim f ()lim + g()lim f() + g() g() g( + ) g() g()g() f ()f () g() f () (ii) g'() lim lim g()lim f ()lim g() f() f() (c) From (), f () g () f(), f() k cos + k sin, f(), f () k, k f() sin From (), g () f () g(), g() k cos + k sin, g(), g () k, k g() cos. 7. f, g r vn f(-) f() nd g(-) g() f(-) + g(-) f() + g() f + g is vn. n Lt () i, tn (-) ( ) i i i i i i (), i is vn. i n i. () Fls, f() is odd, f(). () Tru, () [ g() + g( ) ], (-) [ g( ) + g() ] (), () is vn. n i n i
9. (c) Tru, If f is injctiv, tn f( ) f( ). If f is vn, tn f(-) f(), ut t injctiv proprt, w v -,. Tis lds to contrdiction unlss t domin of f {}. (d) Tru, If f is incrsing, < f( ) < f( ). If >, w v f(-) < f(). If <, w v f(-) > f(). In ot css, w don t v f(-) f() nd f is not vn. () If f, g r incrsing, tn < f( ) < f( ) nd g( ) < g( ). () +, +, Oviousl, f( ) + g( ) < f( ) + g( ). f + g is incrsing. fg is not incrsing, f(), g() r incrsing, ut f()g() is not incrsing. f o g is incrsing sinc < g( ) < g( ) f(g( )) < f(g( )). 6 Intrcpt & Inflion: (,) - - O - - () ' ( 6 + + ) ( + )( ) '' ( + ) Intrcpt : (,) M : (, ),Min:(, ) Inflion :(,), Asmptot : Smmtric out origin. 6 - - - - - O 6 - - - (c) ( ) ', " M (, /) Asmptots : nd Inflion (, /9) - Intrcpt (,) (d) + + () Intrcpt (,),(,-) - - - - O - - - - - ( + )( ) ' ( ) " ( ) Intrcpt (-,) M (-,), Min (, ) Asmptots :, - - - O - - - Oliqu smptot : + Asmptot: - - O 6 - - (f) ( ) ' ( ) ( + ) ( + ) " ( ) ( + ) M Min,, Inflion : (,) Asmptots: - - - - O - - Intrcpt:(,) ±, - Smmtric out origin -
. ( ) ( ) ( ) ' / ( ) ( ) / " 9( ) (-,/) / (/, /) / (/,) (, + ) + + + - ± + - ± + + + + + M :, Point of inflion :. T rquird r A A' ( ), Min : (, ) ±,, < / ( ) /.. - -. - -. O... -. - -. - - - O ( ) A" A" ± ± < - M of A t ±. T rquird distnc D D' ( ) + + (du to smmtr, w tk > for simplicit), ln ln, ln ln + Wn, D. ln ln ln + Wn < <, D < nd wn >, D >. Min of D. -t cot α sin t, -t cot α cos t -t cot α cot α sin t d csc α tcotα (sin αcos t cosα sin t) csc α tcotα sin(t α) csc α d t cot α t cot α csc α[ cos( t + π α) cot α sin( t + π α) ] t cot α ( csc α) [ sin α cos( t + π α) cos αsin( t + π α) ] tcotα tcotα ( csc α) sin( t + π α) ( csc α) sin( t + π α) (n ) π + α sin (t + π - α) t + π - α nπ t wr n Z. tcotα sin( t + π α)
(i) Wn n m (odd), m Z, d (csc α) t t m (m ) π + α..() m cotα t m cot α sin(mπ α) (csc α) sin α < is m wn t t m- s in (). (ii) Wn n m (vn), m Z, d (csc α) t m cot α sin(mπ + π α) is min wn t t m, s in (). (m ) π +..() α t m (csc α) Sustitut t vlus of t m- in -t cot α sin t, w gt : m [(m) π+α]cot α (m) πcot α αcot α { sin α} sin[(m ) π + α] Sinc t, - is dcrsing function, t lrgst mimum is -α cot α sin α, wr t α/ wr m in (). T lst prt grp is sown on t rigt ov. t nvlops wic ncloss t curv -t cot α sin t...() π t ± r. () sin, cos.. () m cot α t cot α sin( π α) (csc α) sin α > t m Sinc cos, trfor is lws twn nd +. () From (), sin sin nπ, wr n Z. nπ /, cos, t nπ / is ± Wn nπ /, nπ /. (nπ /, nπ /), n Z r t points of inflion. B sustitution, points of inflion (nπ /, nπ /) oviousl li on. (c) From (), cos /,, +R, <, < cos < π,. O. -... nπ nπ ± cos, ± cos,. < cos < π nπ For + cos, nπ + cos, sin, ' ' > nπ + cos, nπ + cos r min. points ling on. nπ For cos, nπ cos, sin, ' ' < nπ nπ cos, cos + r m. points ling on +.
(d) (i) For >, tr is no m. nd min. points. 6 (ii) For <, T grp on t rigt wit.7 nd T grp on t rigt. T lins 6 wit. nd. 6 ± r lso drwn wit m. (in ollow dots) nd min. (in O π π π π O π π π π solid dots) sown. -. ( ). Wn,, trfor is vrticl smptot. lim ( ) lim m lim ( ) lim( m) lim ± lim ± lim ± lim ( ) ± ± ± r oliqu mptots. Lt (, k) point on t givn curv. Diffrntit t function, w v d d d d (, k) k Lt m t qution of tngnt wic psss troug t origin. k m k.() k But (, k) is on t curv, w v Solv () nd (), (, k) k, ± ( ).() nd t rquird tngnts r 9 7 ±,or () λ ( ) λ ( ) λ ( ) () () Sinc () is t givn curv nd in ordr tt λ ( ) s tr rl roots s in t grp on t rigt, < λ < /7, compring () nd (). It s tr roots ( -, /, /) if λ /7 nd s onl on root if λ dos not li twn ts limits. - - - - O - - - - 6
. t, t d d d 6t, 6t t d Lt t tngnt(s) to t curv pss troug givn point (, ) t( ). t, t is on tis tngnt. t t(t ) or t t.() () is cuic qution in t nd in gnrl s tr roots. Trfor in gnrl, tr tngnts to t curv pss troug givn point (, ). (i) Considr t function t t.() t t ±.() () s rl solution if >, in otr words, () s turning point(s) if >. Sinc t curv () cuts t -is onc if tr is no turning point nd trfor > is ncssr condition for t tr tngnts to rl. (ii) T cuic qution () s distinct rl roots if nd onl if 7 >, wic is t sufficint conditions for t tr tngnts to rl nd distinct. 6. cos t sin t, cos t sin t. 7. d cos t d [ cos t sin t], sin t[ cos t sin t], [ cos t sin t] d tn t d cos t sin t [ cos t sin t] ( cos t sin t) tn t Eqution of tngnt is givn : cos t sin t cos t sin t cos t sin t( cos t sin t) cos t sin t At, cos t sin t.() tn t cos t sin t cos t ( ) cos t sin t cos t sin t cos t sin t cos t sin t cos t cos t cos t cos cos t sin t t In itr css, Ar / π ( sin t) / or dos not li twn nd d d π / π/ ( cos t sin t) ( sin t) + 6 π, / cost π / π ( sint) t 6 + d d ( + ) ( + ) /. + T function s turning point if + s solution, tt is, [ ( cos t ) ]..... -. -. -. -. -. O..... -. -. -. -. - 7
Δ (-) ()() ( ) or. If, If -, ( + ) ( + ) ( ) ( + ) d ( ), tr is no sign cng out n point on t curv. d + ( ) d ( + ), tr is no sign cng out n point on t curv. d + ( ) In ot css, t function s no m. or min. For <, lt + ( - α) ( - β) α < β. d ( α)( β), nd tr r sign cngs out α nd β. d ( + ) In conclusion, s mimum or minimum vlu if <. Oviousl, + s no solution if Δ < or >. / - O 6 - - - O - - - O. (i) (ii), + d d ( ( + ), ( + ) ) / For t givn sttionr vlu w v : For ts vlus of,, + Wn is sligtl lss tn, <. Wn is sligtl grtr tn, >. (, /) is minimum point., ( ) /,,. d ( ) d ( + ) At t point (, k) on t curv, k,. T grdint from (, ) to (, k) is k/. k/ k / or k. Wn k,, - ( > ) or + ( < ) (rjctd) Wn /, k, (, k) (/, ) m Eq. of tngnt : ( /) or. k () wic s no root if k () < k < < k < - - - O if > if <. - -
9. ( ) f () ( + ) () () f '() ( + )( ) ( + ) () f ''() ( ) ( + ) Wn -,, trfor - is vrticl smptot. () f () ( ) lim lim ( + ) lim + m + ( ) ( ) ( + ) + c lim lim lim lim ( ) ( ) + + ( + ) + is n oliqu smptot of t grp f(). () (i) Wn -, f (), f () -9/6 <. (-, -.) is locl m. point., Wn, f () nd cngs sign cross t vlu, (, ) is point of inflion. (ii) Sustitut in (), ( )( + ) ( ), -/. 6, f() (c) is t t intrsction of t grp nd its oliqu smptot. (d) - - - O - - - - - - O - -. (i) if Countrmpl : f(). Tn f() f() if. But f () is not dfind t. if < (ii) f () f () Tru. Proof: f '() lim >, sinc f() f() nd r of t sm sign. (iii) Tru. Proof: f () > for ll f () f () f '( ξ) >, < ξ <, Mn Vlu Torm. f() > f() wnvr >. (iv) Tru. Proof: f () for ll f () f () f '( ξ), < ξ <, Mn Vlu Torm. f() f() wnvr >. 9