0+150^7^ P.GS - IS PHAN HUY KHiH iL ie uO nT hi Da iH oc 01 / Chuqen de Gia tri I0n nliiK Gia tri nli6 nlid ^Danh clio hoc sinh Idp >BfensoantheonOidun^va w fa ce bo ok c om /g ro up s/ Ta BOI DUONG HOC SINH ww c^utrucd^tliicuaBOGDfiflT • ^ V I E N Tl'NHBiNHT OK] Ha NQI NHA XUAT BAN DAi HQC QUOC GiA HA NQI IJCU N6I N H A X U A T B A N D A I H Q C Q U O C G I A H A NQI 16 Hang Chuoi - Hai Ba Trang - Ha Npi Dien t h o a i : Bien t a p - Che ban: (04) 39714896 Hanh chinh: (04) 39714899; Tong bien tap: (04) 39714897 Fax: (04)39714899 i>Au ww w fa ce bo ok c om /g ro up s/ Ta iL ie uO nT hi Da iH oc 01 / Bai todn tim gid tri U'fn nhd't, nhd nhd't ciia ham so noi rieng vd hat dang thiic ndi chung Id mot nhifng chii de quan vd hu'p dSn tnmg chutfng trinh gidng day vd hoc tap In) mon Todn d nhd trudng phd thong Trong cdc de thi mon Todn ciia cdc ki thi vdo Dai hoc, Cao dang 10 nam gun day (2002 - 2011) cdc hdi todn lien quan den vi^c tim gid tri nhd't, nhd nhd't ciia hdm w thudng xuyen cd mgt vd thut'fng Id mot nhiing cdu kho nhd't ciia de thi , , , , Chiu trdch nhiem xuat ban Vdi li do cdc cud'n sdch chuyen khdo ve chii de ludn luon thu hut su chii y vd Gidm doc • Tong bien tap : T S P H A M T H I T R A M I quan tdm ciia ban doc Tnmg cud'n sdch "Cdc phUtfng phdp gidi todn gid tr\ nhd't, gid tri nho nhd't" nay, chiing toi se cung cap cho ban doc nhvtng cdch gidi thong dung nhd't doi vdi nhiing hdi todn tim gid tri Idn nhd't vd nhd nhd't ciia hdm so.cdng nhu hiet Bien tap vd sita bdi: H A I NHtf cdch dp dung hdi todn de gidi nhieu hdi todn lien quan den no Che ban: Cong ty K H A N G V I E T Noi dung ciia cud'n sdch dUOc trinh hdy chUcfng Trinh bay bia : C o n g ty K H A N G V I E T Chiiong v(H tieu de " Vdi bdi todn md ddu ve gid tri l^n nhd't va nhd nhd't cua ham so" se gidi thi^u vdi ban doc bdi todn tim gid tri Idn nhd't, nhd nhd't ciia hdm sd'thong Chiu trdch nhi^m ngi dung vd ban quyen qua vi^c trinh hdy tinh da dang ciia cdc phUcfng phdp gidi hdi todn ndy Bdng cdch diem Cong ty TNHH MTV DjCH Vy VAN HOA KHANG VI^T lai nhiing sU cd m$t ciia cdc hdi thi ve chii de ndy cd mdt cdc ki thi tuyen nnh Dai hoc - Cao dang cdc ndm tic 2002 den 2011, cdc ban se thd'y duac sU can thie't cua vi$c Tong phdt hdnh: phdi trang hi cho minh nhvtng kien thiic de gidi quyet cdc hdi todn d'y Cud'i chUtfng Id cit sd li thuyet ciia hdi todn tim gid tri Idn nhd't vd nhd nhd't ciia hdm so Phun giup cdc ban nhiing kien thiic chud'n hi can hiet di' doc tiep cdc chUifng sau ciia cud'n sdch C6NG T Y TNHH MTV Cdc phUcfng phdp ca ban vd thong dung nhd't de gidi bdi todn tim gid tri Idn nhd't vd Sm ajP D ! C H vy V A N H A K H A N G V I | T nhd nhdt ciia hdm sd'duac trinh hdy tit chUOng den chuang ., • /^Dia chJ: 71 Dinh T i § n Hoang - P D a Kao - Q.1 - TP.HCM ~ ^ Chitang 2: Phi/mg phdp h&t ding thuCc tim gid tri l^n nhdt vd nho nhdt cua ham sd Dien thoai: 08 39115694 - 39105797 - 39111969 - 39111968 ChiiOng 3: Phiicfng phdp liifng gidc hoa tim gid tri l^n nhdt vd nho nhdt cua hdm Fax: 08 3911 0880 Email: l - ; V - x ^ >0 V x D o f ( x ) > - , V x e ro f(-2) =-2 G j> [-2;2] , [-2; 2] '^'fn - ; (1) ; i, H; :J nm) M (2) T i l f ( l ) ( ) s u y r a m i n f ( x ) = -2 om T a se chu-ng m i n h f(x) < 2V2 c PHANHUYKHAI, V i p n T o a n hoc, 18 DiTcfng Hoang Quoc V i ^ t - Quan C a u G i a y - H a Noi That vay (3) o ok X i n chan c a m dn V x e [-2; 2] X + V - x ^ < lyfl o (3) V4-x^ < > ^ - x ' — - c ^ - x ^ < (2V2-x)^ ( d o x < ) o x ' - 4>/2 x + > bo Tacgia ce « (X - ' 72 )^ > w fa Tur ( ) suy ( ) dung Nhu" v a y ta c f(x) < 2^/2 L a i cd ww - Tacd L a i c6 /g nght rdng chdc chdn cudn sdch ton duc/c tdi hdn nhieu idn) ! s ' H a m so' d a c h o x a c d i n h k h i -2 < x < Mat ddu vc'fi tinh than nghiem tiic, ddy trdch nhi(m viet cudn sdch nhung vdi mot Thtf tCf gop y xin guTi ve theo dia chi sau: ' Nhdn f(V2) = 2N/2 , n e n max f(x) = -2 z^ /: + x y z (x + y + z) x' + y + z" + 24xyz (*) (**) Theo bai tren ta c6: (x + y + z ) ' > x V y ' + z ' + 24xyz Tilf (**), (***) suy ra: P > Bai 15 Cho x, y, z la cac so thifc diftftig T i m gia t r i nho nhat ciaa bieu thuTc +8yz + z' + 24xyz hay (X + y + z ) ' > x ' + Thay (3), (4) vao (2) va c6: P > ro x^+y^+z^+2(xy+yV+zV) 2^x • ' > x ' + y-'+ z ' + 277xyz.\/xVz^ - 3xyz bo hayP> ' (X + y + z)^ = x^ + y ' + z' + 3(x + y + /)(xy + y/ + /x) - 3xyz z^(x^ + xy + y ^ j A p dung bat dang thufc Svac-xd, ta c6: fx^+y^+z^f P> i L x^(y^ + y z + z^) + y^ (z^ + z x + x ^ j + z^(x^ + x y + y^) , (2) - * A p dung ba't dang thiJc Cosi, ta c6: (1) Ta x ^ ^ y ^ + y z + z^j y_ s/ + /g ^1 (x + y + z) x ( x ' + y z ) + y ( y ^ + z x ) + z ( z ^ + x y ) V i e t l a i P dtfdi dang: P= , X/X.N/X^X^ + y z + ^/y.^/y^/y^ +8zx + N/Z.VZI/Z^ + x y z^ z +ZX + X + 8yz + y ^ y ^ + 8zx + z-y/z^ + 8xy A p dung bat dang thiJc Bunhiacopski, ta c6: B a i 14, Cho x, y, z la cac so' thifc di/dng T i m gia tri nho nhat ciaa bieu thtfc 2 z^z^ + x y (x + y + z ) ' ' V a y m i n P = x = y = z = l y-y/y^ + z x TO'(1) va theo bat dang thiJc Svac-xd, ta c6: T i i r ( l ) , ( ) s u y r a P > r (I) P- (2) + y ' + z^ > (x + y + z)^ " \y P > (x + y + z)- 1= x +y + z Vay P = o X = y = z = ^ ^ a i 16 Gia siif x, y, z la ba canh cua mot tam giac c6 chu v i bang 12 ^ I ChuySn BDHSG Toan gia tri I6n nha't va g\& tri nh6 nha't - Phan Huy KhSi Cty TNHH MTV DVVH Khang Vijt Dau bkng Irong (5) xay x = V - x ^ o x = V2 - < F((p) < 2V2 V - | < ( p < ^ , Vay maxy = y/l C:>x = yl2 Cdch 2: (PhiTOng phap chieu bie'n thien ham so) Xet ham so f(x) = x + V4 - x^ vdi - < x < F((p) = - cos v4-x^-x I I S /4-x^ Vay \/4-x' X '5 < 2, ta c6 (4 - x") - x" = - 2x- = — X = N/2 V2 71 CP i max Hx)= max -2 < X < yfl va - 2x' < N/2 < x < 2, nen ta c6 bang 37t • F((p) = 2>/2; ~ ; Tt 7t f(x)= , Cdch 4: (Phu'dng phtip mien gia trj ham so) bien thien sau: ^/2 Gia sur m hi mot gia trj tiiy y cua ham so \'(x) = x + \l4-x' ^^^' Khi phu'dng trinh an x sau day x + \J4-x~ (1) c6 nghiem \ / - x - = m - x (2) Ta Ro rang (1) o =m " i l(x) = min{l'(-2);r(2)) = min(-2;2) = - /g - < ,\ > up max l"(x) = I(N/2) = 2N/2 ; : ro •"''^ s/ B^i loan ltd thanh: Tim m de (2) c6 nghiem Tirdosuyra fa w F((p) = 2sin(p + >/4(l - sin" (p) 2sin(p + c o s ' cp = 2sin(p + 2|eos(p ww TCr ta quy ve xet ham so = 2sin(p + 2cos(p (do - - ^ < cp < ^ ihi coscp > 0) De lha'y y = m - x o x + y = m, y = ^ - x " c^< ^ I y>0 Dc tha'y dieii xay va chi du'cVng thring X + y = m nam giiJa hai du'clng x + y = - va X + y = V2 , ti'fc la vii chi - < m < 2V2 (3) = 2N/2COS((P ) ,^ n n Tir(3)suyra 37t 71 rt Do — < ( p < - => - • — < ( p - - < - 2 4 Tif suy - — ^ < cos f(P ;,i;.f- max r(x) = 2%/2; -2 P = - x = 1; y = up i5«»f^« 'flfffetv''}l>_'n^frflJ j s l i i i f U ' X ^ (y-1)' (1 + y)^ " 4 (2) Tom lai max P = — < = > x = l ; y = 0; minP = ww w Do X > 0, nen liJT (2) suy P > - - V x > 0; y > P = - - o x = 0; y = x = ();y = (x-y)(l-xy) (l + x ) ^ l + y)2 X x = l ; y = 0, d6P = [xy = () x + y = l + xy [x + y = l x = 0;y = l , k h i d6P = - i (1) , Dafu bang (2) xay o xy = 4(l + x ) ^ l + y)- _ 4(l + x)2(l + y)^ : 4y (i + y r 0) Tird6suyraP^^^-y^/^-^^[4 Vx.O;y^O (l + x)^(l + y ) ' Mat khac P = - ' ^ ^ VaymaxP= - x = l ; y = x=l 4 Do vai iro binh dang giffa x va y, nen la co (l + y ) ( l + x)' p _ , U - y K l - x y ) > _ ivx>(),y>0 (l + x)^I + y)^ Mat khac P - - - o x = ( ) ; y = V a y n e n P = Cach 4: (Phifcfng phap lifting giac hoa) o x = ; y = (2) .i '1 i (xem each 1) Ta co: P = Do X > 0; y > 0, ncn hien nhien la c6 x - y | | l - x y | < ( x + y ) ( l + xy) ta co: ( x - y ) ( l - x y ) ^ ( x - y - xy) 2/1 , , \ n j_ v^2/-l v"!^ (l + x)"(l + y ) ' A4(l + x ) ' ( l + y) ( x - y + l - x y ) ^ (l + x ) ( l - y ) ' Lai eo • ' bai loan tren - y - xy (l + x)^(l + y)^ (5) Nhdn xet: Cung suT dung phiTcMg phap ba't dang thiifc, nhu-ng ta co each giai Cach 2: (PhiTdng phap bat dang ihu-c) Ta c6: hay - i < P < | 4 xy = (y-x)(l-yx) ^ fa (1 + x) (1) {\ yf ce P= y ro 4(l + x)2 Do y > 0, nen lij" (1) suy P < - , V ^' , /g ~ s/ (1 + y)^ ^ - i _ (x-ir (1 + y) AB c 4(1+ x)' y om x - ( l + x)' P Ta ok (1 + y)^ X bo (1 + x)^ P = _ CO x = I ; y = 0; minP = — x = 0; y = 4 HuAng ddn giai X (l + x ) ( l + y)^ iL ie uO nT hi Da iH oc 01 / Tir(3) & (4) di den CO nhffng ifu diem rieng cua no •• ' • (3) • Dau bang (4) xay « x + y = + xy Binh ludn: Vdi bai loan 1, la da su* dung bon phu'cfng phap khac de giai •: ^ + y)(l + xy) Mat khac d i lha'y [(x + y) + (1 + xy)]^ > 4(x + y)(l + x y ) va nho nhat noi tren (x-y)(l-xy) (l + x ) ' ( l + y)^ (X i2 [(x + y) + (l + xy)] (1 + x)^ (1 + y)^ Do X > 0; y > 0, nen dat x = tan^a, y = tan^(3, ( ) < a < - ; < p < - - , (1 + tan^ar , ^ ^ " ' f , = lan^acosV - (1 + lan^ p)^ - - < P < 4 L a i l h a y P = ^< P= — « sau d a y ( a n t ) — sin'2p (1) -•^'^ "^'^^ = m t^+2t + Va, pG|();-) nen (2) n sin 2a = sin2p = ()'^ X p =o y = ()' i - * x-O sin2p=I m i n P = - - ^ x = ; y = i*> s/ c u n g siir d u n g phiTdng p h a p ba'l d a n g ihiJc ( b a e a c h n a y l a i k h t i c n h a u ) Q u a , , „,, ro n h o nha't eiia h a m so ddn gidi ' X e t h a i k h a n a n g sau: i om < {i N e u m^l, X - < m < V i i y ( ) CO n g h i c m k h i v a e h i k h i - < m < ,/ / D o m l a g i a t r j t u y y c i i a r(t), n e n t i i " ( ) suy r a ,r Ket hdp , ! ^ Cat7i ; (PhU'dng p h a p m i e n g i i i t r i h a m so) ' D o X ' + y " = 1, n e n l a d a l x = s i n a y = c o s a , v d i a G |(); 271] , KhidoP= 2sin" a + 12sintteosa 1-cos2a+ 6sin2a — = ^ + 2sinacosa +2cos'a sm2a + cos2a + G o i m l a g i a t r i t u y y c i i a P K h i d o phu'dng t r i n h sau d a y ( a n a ) — / \ -2V3 t"+2t +3 , (1 X 1-cos2a + 6sin2a —=m sin a + cos a + ; day I = — va t y e (l) Cdch (3) ( - m ) s i n a - (1 + m ) c o s a = m - m ) " + (1 + m ) ' > ( m - 1)" 3: (PhiTctng p h a p c h i e u b i e n ihiC-n h a m s o ) TacoP^ 2(x^.6xy)^ X * , - cos2a + 6sin2a = m(sin2a + cos2a + 2) -> •., (4) T i r d o suy m a x P = 3, m i n P = - k h i x - + y~ = , , , c ^ m - - m - < ( ) o - < m < ; ^ C O n g h i t M i i D o |sin2a + c o s a | < V2 , V a e |(), 2TC| o , ' V d i d i e u k i e n x ' + y^ = i h i m a x P = 3, m i n P = - Tir (2)o + (4) P = k h i y = 0, l a d i d e n k c l l u a n : (3) CO n g h i e m ( - 2t- + i2t X k h i d o (3) c n g h i e m k h i va c h i k h i A ' > => s i n a + c o s a + > V a |(); 27i| (1) N e u y = ( k h i d o x = 1) L u c n a y P = 2 N c u y ^ K h i d o P = N e u m = , k h i d o ( m - ) ^ 0, nen ( ) c n g h i c m V a y m = la m o t g i a ww x^ + x y + 3y^ ' fa •' w 2(x^+6xy) + y " = 1, n e n ta e o : P = ce 1: ( P h i f d n g p h a p m i e n g i a t r i h a m so) X' c ok (De thi tuyen sink Dai hoc Coo ddn)> khoi B) Hitdng Do +6xy)^ + 2xy + 2y" bo , /g IJai t o a n : G i a silf X, y la hai so ihifc c h o X " + y^ = ':dch up d o la t h a y r o l i n h d a d a n g c u a phiTdng p h a p d u n g d e t i m g i a t r i lofn nha't v a ' (' (3) : m a x P = max r(t) = v a m i n P = m i n r(t) = - y*() ItR y*l) lelR V d i biii loan t r c n la c each g i a i khac nhaii, Irong d o c each T i m g i a t r i UKn nha't v a n h o nha't c i i a b i c u thiJc P = • » ' + 12l = m ( t ' + 2t + ) c:> m ' + m - 18 < o y = i V a y m a x P = ^ < z > x = l ; y = (); ( ) C O n g h i c m D e tha'y v i t ' + l + > V l , t r i c u a h a m so r(t) * a =0 sin a = :' (m - ) t ' + ( m - 6)1 + m = a =— k h i d o phiTcIng t r i n h t^+2l4-3 = sinWos^a-sin'Pcos'P=-sin'2a- TOr(l)suyra luan: • -li—Ili^, G o i m la g i a t r i l u y y c i i a h a m so 1(1) = tan^pcos^p iL ie uO nT hi Da iH oc 01 / K h i P = liinh Cty TNHH MTV DWH Khang Vi^t BDHSG To^n gi^ tr| Idn nha't vA gia trj nh6 nha't - Phan Huy Khii Ta ChuySn j (xemcachl) + 2xy + 3y" N e u y = 0, l h i P = " ; Cty TNHH MTV D W H Khang Vigt Chuy§n dg BDHSG Tpan gi^ tr| Ifln nhat * Ncu y ^ 0, thi P = gia tri nh6 nhat - Phan Huy KhSi ^^^^ vdi t = y Ap dvng (2) vdi a = ^ ; b = f Kb a > 0, b > va ab = - (do x y), r +21 + Dat f(t) = t G R thi f'(t) t^+2l + -81^ + 121 + 36 , ( l ^ + t + 3)^ , 2t^-3t-9 ( t ^ + t + 3)^ ' 1+^ V Ta CO bang bie'n thien sau: -00 ii iH: l ^ ' ^ 1+ z z _ +O0 iL ie uO nT hi Da iH oc 01 / I (4) I_+-L->-^ nen ta c6: Dau bang (4) xay I'd) 1(0 + - Tif (4) ta Tir suy max f(l) = va t'(t) - - teE 'f.f U'W 'n'.l-.U:^:'' Vay maxP = 3, minP = - x^ + y ' = Ta va nho nhat cung the hicn ro qua thi du up 2x + 3y z z+X ro thiJc: P = s/ Bai toan 4: Cho x > y, x > z va x, y, z e [ ; 4].Tim gia tri nho nhat ciia bieu /g (De thi tuyen sink Dai hoc Cao ddiifi khoi A - 2011) om Hii(fng dan gidi c LtJi giiii cua bai toan la su" kel hctp khco Ico cua hai phifcfng phap bat + 3^ ' ' ' ' • X + 1+^ y (1) Xet ham so' f(t) = Taco: f'(t) = ^ + '^ ww Tru"(1c hel ta chiJng minh bat dang thiJc sau: * " ' *' 1 Neu a > 0, b > va ab > 1, Ihi la c6: + > + a + b + x/ab VI t > ^ I (2) f'(t) f(t) Dau bang (2) xay va chi a = b hoac ab = Vay Thatvay(2) > o 1+a l + >/abJ 1^1 + b + Vab Tab (l + aKl + x/ab) (V^-^/b)^^/^-l) >0 (l + a)(l + b)(l + >/ab) Do a > 0, b > 0, ab > 1, vay (3) dung suy (2) dung >yab-b (l + b)(i + (3) 7^) > 7y^ = z x =y A i i ;l I ^ ^ - + - ^ vdi < t < 1+t (31 -61^) + ( ^ - ^ - ^ = (21^+3)^(1 + 1)^ ^ f (t) < V t e [ ; ] TO c6 bang bien thien sau: 1 i • f (t) = f (2) = | TO suy P > ^ , 33 l - ^ + 7~T3 1+t 21^+3 l + l 2+ w • •+ bo P diTdi dang: P = ce Ihtfc 1+ 21+3 61 fa Viet lai bieu (5) X Datt = E D o x > y v a x y e l l ; ] n e n s u y r a l < ^ < = ^ l < t < K h i d ok dang ihiJc va chieu bien thien ham so nhif sau: x=y + Dau bang (5)xayra Binh luqn: Tinh da dang cua cac phU'dng phap giai bai loan lim gia tri Idn nhat y y+z P> X • ' =z y y 2+3^ leM x CO ^ y X ^ —•— Chuygn dg BDHSG Toan gii trj I6n nha't va glA tr| nh6 nha't - Phan Huy KhAi - Phiftlng phap mien gia tri ham so 'i ^ j^ifif'^f^'M^^'•''•>::^r^' - Phi/dng phap lU'dng giac hoa c.^i '.-rl- Phi/ctng phap hinh hoc hoa ,;:^^^iry, h^> rti - Cac ban cung da tha'y dtfdc chiing ta c6 the c6 nhieu phU'dng phap khac de giai cung mot bai toan tim gia tri Idn nhat va nho nhat cua ham Do X, y, z e [ 1; 4] ncn P = — x = 4, y = l , z = 33 34 Nhu" the minP = x = 4;y = l;z = 33 ; ~" Bai toan 5: Cho bon so ihifc a, b, c, d thoa man dieu kien a^ + b" = c' + d"^ = Tim gia trj Idn nha'l cua bieu thiJc P = > y - a - b + V - c - d +N/5 - a c - bd Hii(fng dan gidi Ldi giai hay nhat va dac sSc nhat cho bai loan la phu^dng phap su* dung hinh hoc sau day: Ta thay cac diem M(a; b), N(c; d) va Q( 1; 2) a, b, c, d la cac so thifc thoa man dieu kien dau bai deu nam tren difdng Iron c6 tam tai go'c toa va ban kinh bhng v ' f i"i: V ^^ Viet lai bieu iMc P dxidi dang sau: f ? x^vlfi^m/(a-l)2+(b-2)^ ka-cf+ih-df P= iL ie uO nT hi Da iH oc 01 / SINH V A O D A I H Q C , C A O D A N G v2 = -N/30 Qua bai toan tren, chung toi da gidi thieu vdi c&c ban cac phiTdng phap? chinh de giai bai toan tim gia tri Idn nhat va nho nhat cua ham so" - PhiTdng phap bat d^ng thtfc - PhiTcfng phap chieu bie'n Ihien ham so Cac bai toan tim gia trj Idn nhat va nho nha't cua ham so thu'dng xuyen xua't hien cac ki thi tuyen sinh vao Dai hoc, Cao dang nhiTng nam gan day Trong muc chung toi xin gidi thieu cac bai toan ay kem theo nhffng binh luan can thiet , ,/ • ^ v Bai 1: (De thi tuyen sinh Dai hoc Cao dunf- khoi A-2011) Cho X , y, z la cac so thiTc cho x > y, x > z va x, y, z e [1; 4) Tim gia trj ^ t X y z nho nha't cua bieu thuTc: P = + + "• :: - Ta s/ /g om c ok bo ce fa w P < ^ ^ ww 3^30 ro d day CMNQ la chu vi cua tam giac MNQ Ta sur dung ke't qua quen bict hinh hoc phiing sau day: Trong cac tam giac npi tiep mot di/dng tron ban kinh R cho trU'dc, thi tam giac deu la tam giac CO chu vi Idn nhat Mat khac tam giac deu noi tiep du'dng tron c6 ban kinh R c6 canh b^ng R ^/3 Do d6 maxP = N H O N H A T C U A H A M S O T R O N G C A C KJ THI T U Y E N up (MQ + NQ + MN) = VivayCMNQ< N / l T i r ( l ) s u y r a §2 N H I N LAI C A C BAI T O A N V E GIA TRj L N N H A T V A 2x + 3y y+ z z + x • l; ' 1^)'?' HUdng ddn gidi Xem Idi giai bai toan 4, muc §1, chu'dng cuon sach Binh luan: Mau chot de giai bai la d cho bang each su" dung mot ba't dang thiJc phu, de diTa ve danh gia P > — ^ — + — y (1) De'n day b^ng each difa vao an phu t = ^ -• vdi t G 11; 2] ta quy v^ danh y gii P > + _ L f(t) '^H*>v^^-; • 2t2+3 + t R6 rang tiep theo ta nghT den se sur dung phiTdng phap chieu bien thien ham so de tim f(t) vdi < t < Tir ke't hdp hai qua trinh tren ta se di den IcJi giai cho bai toan Van d6 la d cho viec phat hien (1) khong phai la dieu de dang Chuy6n dg BDHSG Toan gii trj I6n nha't va gi^ tri nh6 nha't - Phan Huy Kh5i Cty TNHH MTV DWH Khang Vigt Thay cho vice suT dung mot bat dang thtfe phu, ta co each lam sau day c6 ve "tiT nhien " hcJn mot chiit Coi P nhiT la mot ham so'eua z, xet ham so'an z ' P = P(z)= — + - ^ + - ^ vdiz G [l;x] '>;< 2x + y y + z z + x Khido P'(z) = () L + (y + z)^ {z +X xf _ x(y+ {y +z)^-y(x zfiz + +xfz)^ _ Tim gia tri nho nha't cua bieu thtfc P = (1), * N e u x ^ y , lhiP(z)= ^ + ^ ^ + - ^ = ^- V z e [ l ; x ] 5y y + z z + y - i * Ne'u X > y (chii y la x > y, nen x y thi x > y) thi x - y > nen ii^.'"' P'(z) = 0z^-xy = 0z= ^xy • aiaj',;;:.;J.J:-:^'^ Ta CO bang bien thien sau (suy tif (1)) ? , ! ' !7 z P'(z) + P(z) ^ - ^ ^ — ^ Hi!K i Da'u bang (2) xay o z = ^xy s/ up ro om ok fa ce bo ^ slx+yfy (2) w p ( z ) > ^ - ^ ^ Jy 2x + 3y + P = P(z)> — -1— + —2 c X y 2x + 3y " y + ^ " 7^ + x ww p(7^)= /g Vay vdi mpi z e [1; x], ta c6: P(z) > Hiidng dan giai a b\ f a h\ a b^a b — — -3 Difa P ve dang sau: P = Vb— + &) + b + a— - h— b a b a\ a b^ a b a b = — + — + b ^ a j - —b + —a 18 (1) • b a j Viet lai gia thie't diTdi dang sau: r a b ' +1 = (a + b) + b aJ ab '2,\/^ Theo baft dang thtfc Cosi, ta c6 I + — > - p = ab vab ^a b " + l > ^ ( a + b) = 2V2 — Thay (3) vao (2) va c6: vu + ay Vb ^ Va (4) Vab b a Ta r — b^^ + b^ a^ + ) iL ie uO nT hi Da iH oc 01 / (x-y)(z^ - x y ) {y + zf(z + xf Bai 2: (De thi tuyen sink Dai hoc Cao ddrtfi khoi 8-2011) Cho a, b la hai so thiTc dUdng thoa man dieu kien: 2(a^ + b^) + ab = (a + b)(ab + 2) 34 Den day ta lie'p tuc giai nhiT phan sau cua bai 4, muc §1 v6i lifu y rkng - >— 33 34 nen minP = — 33 Ro rang viec phat hien (2) theo cdch giai "tif nhien" hdn each giai cua bai 4, mat du no phuTc tap hcfn ve mat tinh toan! Khi tir(4) ta c6 2(t^ - 2) + > > ^ t hay 2t' - V2 t - > =>(>/2t+l)(N/2t-3)>0 Do t > => t + > 0, nen tiif (5) suy ' (5) V2t-3>0=>t>^=>i + ^=t^-2>l (6) ^/2 b a Bai toan quy ve: •>^'-*v-'.^*'.'^^ Tim gia tri nho nha't cua ham so f(t) = 4t^ + 9t^ - 12t - 18 vdi t > - Ta c6: f'(t) = 12t^ + 18t - = 6(2t^ + 3t - 2) va c6 bang xet dau sau: -2 t 2 f'(t) 0 + + + f(t) 23 23 (7) Vay minf(t) = f = - —4 t t f c m t a c d P > - — '4 i 15 Cty TNHH MTV DWH Khang Vi$t Chuy6n dg BDHSG Jo&n gii tr| I6n nhflt 1 O 1 - + - = — + — X g\i trj nh6 nha't - Phan Huy Khi\ 2y y ' xy (1) ^ • i D a t X - - ; Y = - k h i (1) c6 dang: X + Y = X y - XY + YI f'(y)- (2) V d i dieu k i e n (2) t h i A = ( X + Y ) l Vl2y> T i m gia trj Idn nha't cua A = (X + Y)^ v d i dieu k i e n (2) X + Y = (X + Y)^ - X Y ' ;„ ^4J;' ^ y r ( X + Y)^ nen tuf (4) ta c6 (X + Y)^ - 4(X + Y ) < o V a y t a c o A < 16 - • (5) I' ' E^-f, +7(x + l ) ^ + y ^ + y - c HUbng dan gidi Theo phep tinh ve vectd ta c6: ok L a y i i = (x - ; y ) , V = ( - x - ; y) => u + v = (-2; y ) ' w Da'u bang (1) xay u, v la hai vectd cung phi/dng, ciing chieu ww = - X - X = T i r ( l ) s u y A > 2^1+ y^ +|y-2| 9fi up Nhi/vayA>2+ N/3VX, y G R v a A 2^/r+y^ + y - neuy>2 2sjl + y^ + - y neuy- Tilf ta CO mina = + N/3X = 0, y = ^ Hi/dngdan x21nx.i-ln'x Tac6 f'(x) = vdiyeR E x = • ' Dau b^ng (2) xay X = X e t h a m s o f(y) = l + y^ + y - Vye B a i : (De thi tuyen sinh Dai hoc va Cao ddnfi khoi B) ce (1) fa + l)2+y2 > ^ ^ ^ V3 phdp chieu b i e n thien h a m so de g i a i tiep bai toan dat bo u + V hay '=2+ ro ? I • om V ( x - l ^^+y^ )' ' /g Cho X, y e R T i m gia tri nho nhat cua bieu thuTc: Ta CO f(y) = + + Ta •vj ^., B a i : (De thi tuyen sinh Dai hoc Cao ddnjf khoi B) X - TuT suy f ( y ) > f s/ de giai bai toan o ^/^7ne'u:^ + y^ L u c A = X ' + Y ' = ( X + Y)(X^ - X Y + Y^) Chu y rang (2) o 2y + giai • 21nx-ln'x lnx(2-lnx) x^ X' D o x^ > V X e [ ; e^ nen ta c6 bdng bie'n thien sau (di/a vao tinh dong b i d n cua h a m so y = Inx k h i X > 0) Cty TNHH MTV DVVH Khang Vi?t X Inx i i - Inx + y' + _ + - L a i CO ! ' ( - ) = Tu" suy — Vcjy max R x ) - ("(c") = 4"; niin !(x) - • (X + 1)" X- + l + 2x , 2x Laiihay r ( x ) =^ = = \ —rx- + X- + x^+1 r 0-^ > ( ) V x e - ; 2| , 2x Dol < f ' ( x ) < =5 l"{x) < N/2 , V x - ; 2| iL ie uO nT hi Da iH oc 01 / -' y + x+ D o x > - l = ^ x + >()=>r(x) = x^ + M a i khae ( ) = %/2 , iCr suy l"(x) = m i n i 1(0); r(c^)} = m i n j o ; ^1 = max H\) = \f2 -I3N/2 +4 • = 2V2.(5) V x2+y2 (6) Dau bing (6) xay dong Ihdi c6 dau bing (2), (3), (5) fa Nhir vay dinh nghTa gia tri Idn nhat va nho nha't deu c6 hai phan Can liAi y (1) yj Ta Ton Tac6: x + — >>/2; y + — 2x 2y s/ , X y 1' f ' x + — + y+zr- + - + - +— — + y ; ly ^ ) U 2x> [ w r i n g ca hai phan deu quan nhu-nhau, khong diTdc xem nhe phan o x =y= ww Xet thi du sau day: Cho X > 0, y > va x^ + y^ = Tim gid tri nho nhat cua hiiu thtfc sau: Nhir vSy ton tai (x,,; y«) thoa man xf, +y?) = va S = 3^12+4 x = x,,; P = (l +x) y = y,) Theo dinh nghTa ve gia tri nho nhat, ta c6 minS = 3yl2 + + (l +y) yj Xet phep giai sau day: P = + f X I Qua thi du ta thay neu khong de y den di^u kiC*n dinh nghTa gid 1^ + x; + — f y + 1^ - + k X y — + - tri Idn nhat v^ nho nha't cua ham so c6 the se dan den sai lam y) Taco x + - > ; y + - > v a - + ^ > X y y X 31 Cty TNHH MTV D W H Khang Vigt Chuy6n dg BDHSG ToAn gia tr| Mn nhift vji glii tr| nh6 nha't - Phan Huy KhJi B Cdc tinh chat cua gid tri l^n nhat vd nho nha't cua ham so f(x) = min f(x); f(x)|> Gia sur ton tai max f(x); max f(x); f(x); f(x) xeA Khi ta CO xeB xeA i : > > ;• max f(x) < max f ( x ) , (1) f ( x ) > m i n f ( x ) (2) xeA xeB xeA Chtfngminh: v ; xeB xeB suf max xeD f(x) = f ( X ( , ) , vdi x,, e A Do X(, e A ma A c xeB • •'" « ' ' toSn tifdng lir) V i Di c D, I = 1, nen theo tinh chat 2, ta c6: xeDi X(, e Gia t •" * < , ; xeD xeD Chi?ng minh: Gia suf max g(x) = g ( X ( , ) , vdi x„ e D Ta CO f(x) > g(x) V X e D =^ f(xo) V', v up Do max f(x) > f(x,,) > g(x„) = max g(x) => dpcm xeD ro xeD c thi chi/a the kct luan diTdc f(x) > g(x) V x e D xeD xeD] xeDi ww Xet mot vi du minh hoc sau day: D2 Chox>0, y>Ova x + y < t^ Tim gia tri Idn nha't cua bieu thiJc P = x^y(4 - x - y) DatD= {(x;y):x>0;y>0; x + y ; y > ; < x + y < } , D2= {(x;y):x>0;y>0;x + y 0; y > nen P < V (x; y) € D, L a i c < ( ; ) G D| va k h i d P = 0, ncn max P = iL ie uO nT hi Da iH oc 01 / ^+-^- + y + ( - x - y ) y =I (do (x; y) , ro xeD ChuTng m i n h : Gia suT M = max f ( x ) xeD xeD , om xeD c xeD /g f ( x ) K h i ta c6: max f ( x ) = - m i n ( - f ( x ) ) ; f ( x ) = - m a x ( - f ( x ) ) ok X6D max Q = max{();64) = (x;y)€D Bay gid tCr (1), (4) suy , (4) , I mm P = - O.I' (x;y)€D T i n h cha't 5: Cho cac h a m so l"i(x), i'2(x), f„(x) cung xac dinh tren m i e n D D a t t ( x ) = l|(x) + t2(x) + + l„(x) ' Gia stir ton tai m a x f ( x ) , m i n f ( x ) , m a x f j ( x ) , m i n f i ( x ) X6D xeD xeD xeD vdi moi i = i n K h i ta c6: m a x f ( x ) < m a x f , ( x ) + maxf2(x) + + m a x f „ ( x ) , xeD xeD xeD xeD (1) xeD xeD (2) ( NhiT vay ta d i den max f ( x ) = - m i n ( - f ( x ) ) =>dpcm , maxi;(x) = fi(x„),Vi = l n , Phan sau chiJng m i n h hoan toan tiTctng tiT i ' Nhan xet: T i n h cha't cho phep ta chuycn bai toan t i m gia tri Idn nhat bai todn t i m gia t r i nho nhat hoSc ngmJc l a i D i e u nhy c6 ich nhieu tri/dng hdp cu the se xet sau X t t h i du m i n h hoa sau day: 1A max Q = 64 (3) Da'u b^ng (1) xay k h i va chi ton tai x„ G D cho xeD ' ' * D, (x;y)eD| Theo dinh nghia cua gia t r i nho nhat, tCf (*) suy m i n ( - f ( x ) ) - -M xeD X + y - < 4) xeD fa VxeD ww TCf he tren suy ce bo ;M V x e D K h i theo dinh nghla gia t r i Idn nhaft, ta c6: • ^^^^ ^ [f(x„) = M,vdix„eD f-f(x)>-M G D, x +y - V (x; y) e Ta , s/ max P - m a x { ; } = (x;y)eD Tinh cha't 4: Gia suf h a m so f ( x ) xac dinh tren D va t o n t a i max f ( x ) va xeD hay Q < Mat khac (4; 2) G D , va Q = 64, nen ta c6 (3) up Tir(l)(2)(3)suyra (2) -]2 Q= - - y ( x + y - ) < x =2 max P = max Q = (x;y)eD2 ^- + ^ + y + (x - y - 4) X X h a y P < V ( x ; y ) e D2 (x;y)eD2 (1) K h i (x; y) G D | => x + y - > 0, nen theo ba't dang tMc Cosi, ta c : ^4 Ro r a n g ( ; 1) e D2=> ' (x,y)€D Mat khac (2; 2) G D2 va Q = nen ta c6 V d i m o i (x; y) e D2 thi - x - y > 0, nen theo ba't dang thufc Cosi, ta c6: M a t khac lit — - y = - x - y < = > m i n P = - max ( - P ) (x.y)eD Ta CO - P = Q = xV(x + y - 4) K h i (x; y) e D2 =^ X + y - < => Q < (2) {x;y)eD| P = 4||y(4-x-y)^ j l .sm''2x + = - -sin^ 2x + — ^ — +4 cos''xy sin xcos x xeD 'v^^ xeD Da'u b a n g (1) xtiy k h i va chi ton tai x,)e D cho minri(x) = ("((x,)), V i = l , n , , ; xeD Ti/dng tirdau bang Irong (2) xay va chi ton lai x,, e D cho niini;(x) = i ; ( X o ) V i = U , ' • xeD Chu-ng m i n h h o a n l o a n lifdng liT nhiT chi'fng minh ciia tinh chat h(x) = f(x) - g(x) Gia stir ton tai cac gia Irj Idn nhat, nho nha't c i i a cac h a m so f(x), g(x), h(x) trcn D Khi ta cd: m a x h(x) < m a x l"(x) - m i n g ( x ) , (1) m i n h(x) > f(x) - m a x g(x) (2) (1) - ' sin^ 2x XGD xeD Dau bhng ( ) x a y va chi ton tai x„ e D cho maxiXx)=:l"(Xn);ming(x) = g(x„), xeD vdi g ( x ) = — s i n ' ' x ; h ( x ) = 16 (2) niinl„(x) xeD J - sm 2x =5 sin22x + 16—^= + g(x) + h ( x ) , , sin^2x (1) max r,(n) Tinh cha't 7: Gia siif l"(x) va g(x) la hai h a m so c u n g x a c dinh i r c n m i e n D Dill ww Viet lai f(x) dxid'i dang sau / I fa w sm" x + ce Tim gia tri nho nha't ciia ham so + om thiTc hien diTdc neu nhiT cac trifrJng hdp ton tai mot diem x,, ncn cac Xet thi du minh hoa sau day: D X6 Dat f(x) = fi(x)r:(x) l'„(x) Khi la c6: Phan dao chiJng minh hoan loan imtng lif va xin danh cho ban doc ham phan ciing d;U gia trj Idn nhat (nho nha't) tai diem a'y xeD maxf„(x) Cling nhu" minl'ifx), minf„(n) f Tiif (5), (7) suy triTdng hdp xay dau bang ( I ) l(x) = cos' X /(•A va h(x„) = (thi du chon x„ = ttfc la ton tai x„ ma g(x„) = mini g(x); h(x„) = h(x) iL ie uO nT hi Da iH oc 01 / X6D (2) DC-thay ming(x) = - ^ ; h(x) = 16.^ = xeD • xeD Dau b a n g (2) x a y k h i va chi Ion tai x„ e D cho sin" 2x 37 Cty TNHH M T V DVVH Khang Vjgt^ Chuyfin 66 BDHSG Toan gia t r i Idn nha't va gia trj nh6 nha't - Phan Huy K h i i minf(x) = f(X(,);maxg(x) = g(x,)), xeD ; xeD •', Neu them vao gia thie't f(x) > V x e D Khi vdi moi n nguyen dtfdng, , ta co: max f(x) = 2n/max f^"(x); m i n f ( x ) = 2n/min f^"(x) C/ji^n^'m//?/j.-Ta chi can chi^ng minh (1) xeD Ta CO h(x) = f(x) - g(x) = l(x) + (-g(x)) Theo tinh chat ta c6: maxh(x) V X e D, ihi max f(x) = /minf^(x); f(x) = /minf^(x) Van theo tinh chat thi dau bang (3) xay va chi ton tai x,, e xeD D s a o c h o t a c o : maxf(x) = f(x„); max(-g(x)) =-gCx,,) yueD xeD ^xeD Dieu ra't c6 ich de giai cac bai toan thuoc dang Tim gia tri Idn nhat, Nhi/ng max(-g(x)) = - g ( x „ ) o - ming(x) = - g ( x „ ) ming(x) = gCx,,) xeD xeD nho nha't cua cac ham so f(x) chiing diTdc cho di/di dang can bac hai hoac xeD => la dpcm ' CO chtfa cdc bieu thiJc vdi da'u gia tri tuyet do'i ' Xet thi du minh hoa sau day: TiTdng tur ta c6 tinh cha't sau (vdi each chu'ng minh hoan toan tu'dng tiT) ' ' ' Tim gia tri Idn nha't va nho nha't cua ham so': Ta Gia sijf f(x), g(x) la cac ham so xac dinh va di/dng x e D Dat h(x) = • ro j , /g maxlXx) maxh(x)/2)t + - V " Gia siuf f(x) la ham so xac djnh tren mien D Khi vdi moi n nguyen difdng, ta co , , F'(t) V2+I - F(t) maxf(x) = 2n+i/max(r^"^'(x)); minf(x) = 2n+i/min(f^"^'(x)) 38 xeD V^eR , xeD \xeDV , _ ^ (x/2+l)t + 2+V2 F'(t) TinhchaftS: , ,,, ww xeD ,., m a x f ( x ) - /maxf^(x); f(x) = /min f ^ ( x ) Dau b^ng (1) xay va chi ton tai Xo = D, cho , X e t h a m s o F ( t ) = 2t + t + l + t + -^—!- = + t + V V t ^ + t + l - (.v,/,VK):iv xeD Do f(x) > V X e R , nen theo tinh chat ta co """" ' |« ' ' ' bo (2) * '" ok c xeD minf(x) minh(x)>^^-^^^ ' maxg(x) , s/ f(x) = V l + sinx + V l + cosx , x e M "Iv.smy'^'* + Vay m a x f ( x ) = /max F(t) = , / m a x ( ( F ( - ^ ) ; F ( > ^ ) xeR ^\\\ Khong giam tdng quat c6 the cho la M > m > (neu > M ton xeD Ta om /g xeD ro max f(x), f(x) Khi ta c6 tai up tren D va max f(x)| = max < max f(x) f(x) xeD (1) c xeD ok xeD bo ChiTng minh: Ap dung tinh cha't (4), thi he thiJc (1) co dang tu-dng diTdng sau: f(x) = max ( max ff(x) max |f(x)| ( x ) ; max (-f(x)) xeD xeD xeD = max max f(x) max (-f(x)) xeD xeD Khi 66 ta c6: f(x) = (min f(x); max f(x) I xeD (xeD) xeD2 Theo gia thiet ton tai |f(x) va xeD| mm f(x) = f ( x ) , • w |f(X().)| = - f ( X o ) • - c6: f(x) max (-r(x)) xeD ; : ; • r • xeD2 " 02 xeD , (2) hay - f ( x ) > l ( x ) , V x G D > max f(x) xeD2 Mat khac, gia s\{ max f(x) = f(X(,), x„ G D2 nen ta c6 |f(X()) xeD2 , V X G D2 max f(x) (4) xeD2 Tijf (3), (4) va theo dinh nghla gia trj nho nhat cua ham so, ta thu di/dc X6D2^ < max (-f(x)) < max(-f(x)) , ^• (5) (l"(x)) = max f(x) ^ xeD V i vay ta cung c6 |f(x„ )| < max max (-f(x)) xeD max f(x) xeD Tif (4), (5) va de y rang x„ la phan tuf tijy y cua D, suy 40 I - (4) i (1) i ( Lai theo gia thiet thi ton tai max f(x), ttJc la f ( x ) < max f(x), V x e D2, f(x) xeD ww Ne'u 1"(X()) < Luc lai 1max xeD xeD| xeD] Neu f(x„) > Khi la c6 |f(x,)) = f(x„) < max f(x) < max f(x) (3) f(x) < max ( Chtfng minh: V i D, = {x G D I f(x) > 0) nen |f(x)| - f ( x ) , V x G D, Lay y x„ e D, xay hai kha nang sau: xeD (9) xeD fa iI xeD ce xeD "eD (2) i xeD D, = {x G D : f(x) > } ; D2 = {x G D: f(x) < ) s/ xeD dinh (8) :v Tinh cha't 11: Xct ham so l(x) vdi x G D va gia suf ton tai f(x), max l"(x), > m chiJng minh tufdng tu") Do 1(1) > Tuf ta c6: ^ mm f(x)| = m i n f ( x ) = m = min{M,m) = m i n { | M | , | m | | =>d6 la dpcm xac tiTdng TH (6), (9) va theo dinh nghla gia tri I6n nha't cua ham so, ta c6 xeD so U :}Ju f(X„),X„GD tokn Tir (2), (3) suy f(Xo) IXx,,) = max( max f(x) max (-f(x)) •f ham (7) xcD xeD Difa vao tinh lien tuc cua f(x) suy ton tai Xo G D ma f(X()) = suT f(x) la xeD Gia suf max f(x) • ' 10: Gia max Hx) max (-f(x)) iL ie uO nT hi Da iH oc 01 / , ,, Neu M m < 0, ta c6 M > > m Tinh cha't (6) xeD (TriT^ng hdp ngiTdc lai diTdc chufng minh bang mot each hoan [min{|M|,|m|}, neu Mm > ChiJng minh: max (-f(x)) I Khong giam tdng quat c6 the cho la: tifdng (jTng la gia trj Idn nhat, nho nhat cua ham so' l"(x) tren mien D, thi: „ xeD i^>:jiU Tinh cha't 9: Gia suT l(x) la ham so xac djnh va lien tuc tren D Khi neu goi M , m [O.neu Mm < max f(x) < max (5) xeD2 Ap dung nguycn l i phan (tinh chat 4) ket hdp vdi (1), (5) ta c6 max f(x) f(x) = mm f(x); f(x) = (min f(x); xeD2 xeD • dpcm xeDi xeD2 ) ("eDi ChuySn BDHSG Join g\& trj I6n nha't aAut^^2 Cty TNHH MTV DWH Khang Vi^t gia trj nh6 nhS't - Phan Huy Khai PHIiaNGPHllPSADMNG + l ) + (y + l ) + ( z + I ) ] BifrflKiiaT H O C of T I M GUi T R I I H N N H J T T lANHiNHAtcOAHAMSdf (4) iL ie uO nT hi Da iH oc 01 / Mat khac da'u b^ng (3) (ttfc la (2) xay xac dinh gia tri Idn nha't (hoac gia trj nho nha't) cua ham so f(x) Sau ta ^ o x =y= z=- [x + y + z = l ,,3 CO nhiem vu la chi da'u bang xay nc'u nhiTchon du'dc Xo e D de c6 (fJ 11 (3) Ttf (1) (3) suy r a P < - dang thijrc f(x) < M hoac f(x) > m V x e D, d day D la mien ma tren ta can ?I PhiTcJng phap suT dung triTc tiep djnh nghia cua gia tri Idn nha't va nho f(x„) = M hoac f^x„) = m (2) >9 x+1 iIrl: Vay da'u b^ng (4) xay o x = y = z = - V i CO ra't nhieu phiTdng phap de chrfng minh ba't dang ihufc (nhiT suT dung bai dang thifc Cosi, bat dang thuTc Bunhiacopski, phi/dng phap xuat phat tijr cac Tir d6 ta c6 maxP =' - o x = y = z = - ba't dang thtfc da bie't, phu'dng phap nhom hoac them bdt cac so hang ) Trong chufdng chung ta se Ian liTdt xet cac phiTdng phap cd ban nha't silf Bai2:Chox>0,y>0va ^ +~ + s/ Tim gia tri Idn nha't cua bieu thtfc P = — up ro 1.1 SiJ dung bat dang thtfc Cosi cd ban Ta dung bat dang thiJc de giai bai toan dat §1 PHaONG PHAP SUf DMNG BAT DANG THLfC COSI j , >, +V^^z ^ Hudng ddn gidi Ap dung ba't dang thuTc Cosi cd ban hai Ian lien tiep, ta c6 ban: < i — + — — + - —+ — 2x + y + z 2x y + z j 2x"^4 /g Ta goi hai bat dang thuTc thong dyng sau day la cac bat dang iMc Cosi cd om '•' ok c n > vdi moi a > 0, b > Da'u bang xay o a = b (a + b) fi - +— a b 1 n (a + b + c) > vdi mpi a > 0, b > 0, c > -+—+iHffl U b cj Da'u bang xay o a = b = c 'JT ce bo •J' 1 ^1 - +— + — < — 2x + y + z X 2y 2z) w fa Da'u hlng (1) xay o suT dung hai bat dang thuTc noi tren Bai 1: Cho x > 0, y > 0, z > va X ww Ra't nhieu bai toan tim gia tri Idn nha't va nho nha't cua ham so quy ve viec Tim gia tri Idn nha't cua bieu thiJc: P = X + o x = y = z •< — 2y + z 2x y y+1 2z; 1 — + — +x + y + 2z 2x 2y z j Da'u blng (2), (3) deu xay o x = y = z z+1 Hudng ddn gidi (2) (3) Viet lai P difdi dang sau: 1 + 1-= 3y+1 z+1 ^ r -+ • x+1 P= 1-+1 x+1 r''_ L i luan tiTdng tir, c6 + y + z = (1) C n g t i r n g v e ' ( l ) ( ) ( ) v a c6 P < ^ 1 + x+1 Ap dung ba't dang thtfc Cosi cd ban, ta c6: y+1 -f z+1 (1) Do i + i + i = ^ P < i X y z 4 -+— +• 2x 2y z j (4) ^ ' Chuygn de BDHSG To^n gi^ tr| Icin nhS't Cty TNHH MTV DWH Khang Vigt gJA tr| nh6 nhat - Phan Huy Kh5i HUdng ddn gidi Da'u bang (4) xay dong thtJi c6 dau bang (1) (2) (3) •\ 1 ' x = y = z = — ( k c l bdp v6i d i c u k i c n — + — + - = ) X y z Vie't l a i b i c u thtfc P di/di dang: ^^ 2x + y + z Tur suy maxP = l < = > x = y = z = ^ = - ( X + y + z) Nhdn xet: ThiTc chat bai toan l u y c n sinh D a i hoc Cao d^ng k h o i A - 2005 c6 y2 Tim gia t r i nho nhal ciia bicu ihi'rc P = +——+ +x +y 1- x 1- y X+y 1-y + x+y , >9i •^.•(Ms:>.' > - 2' 'i;''?:^'Oit^ ;j T i r ( l ) ( ) s u y r a P > | Chu y rkng k h i x = y = j w =x+y o x =y= j t h i x + y < va liic P = ^ V a y minP = ^ • C i d t r i nho n h a l dat du'dc k h i va chi x = y = - B a i 4: Cho ba so dudng x, y , z thoa man d i c u k i e n x + y + z = 2x + y + z 44 ^.'•'••''''•^••v> ^•^;•r'••' X + 2y + z -+• ill ' i.-: 1 ^ • + y + 2z; 2x + y + z- + x + 2y + z + x x + y + 2z up ^ , |^ K c l hdp v d i X + y + z = 3, suy P = - o X = y = z = V a y maxP = - Gia t r i n^y dat di/dc k h i va chi k h i x = y = z = Bai 5: Cho x > , y > O v a x + y = - HUdng ddn gidi - 1 V i e t l a i b i e u thuTc P di/di dang sau: P = - + - + — (1) X X 3y 1 A p dung bat dang thiJc Cosi cd ban, ta c6 - + — > (2) X 3y x + 3y D a u bang ( ) xay o x = 3y • L a i theo bat d^ng lhi?c Cosi cd ban, la c6: T i m gia i r i idn nhat ciia b i c u thuTc P - + x- + y + 2z T i m gia t r i nho nhat cua b i e u thtfc P = - + ^ x 3y (2) ww D a u bkng (2) xay r a < = > l - x = l - y x + y + 2z (3) Ta ' om I - X > 0, - y > bo x+y (1) ce 1-y fa 1-x •+ + —!^ ro 1-x Theo baft dang ihifc Cosi cd ban, ta CO 1 [ ( l - x ) + ( l - y ) + (x + y ) } + + • 1-x 1-y x + y + — c D o X > 0, y > va X + y < x+y — /g i - y = ok 1-x s/ V i c t l a i P d i T d i dang I •+- (1) Tird6thayvao(l)vac6P 0, y > 0, z > va — + — + - = X y z Chtfng minh bat dang thiifc x+y+ z 4 / 1 — + =4 — + 2x' x + 3y 1^2x x + y j 4 ^ 16 hay — + —> 2x x + 3y 3x + 3y - v d i m p i a, b, c > b+c c+a a+b ^ ; w = J — K h i u > 0, v > 0, w > Vxy V zx L u c n ^ y (1) c6 dang u^ + v^ + w ' = (1) wu + vw ww x+1 + Hudng ddn gidi + ^ fa + w T i m gia Irj nho nhat cua bieu thiJc P = P= = (uv + v w + wu) B a i 9: Gia suf x, y, z la cac so thiTc diTdng va th6a man dieu k i e n x^ + y^ + z^ = 3xyz I la cdc so difdng (Wc 66 TO rang x > 0, y > 0, z > va xyz = 1) s/ y+z x = y= z= L U V w Do xyz = 1, nen thuTc hien phep d o i bie'n sau: x = - ; y = — ; z = — v d i u , v, w V w u up x+y >• ' T i m gia tri nho nhat cua bieu thtfc P = — ^ — + — ^ — + — ^ — xy + x yz + y zx + z x + 2y + z Cong tuTng ve' ba ba't dang thtfc tren r o i rut gon, ta c6: + + uv Bai 10: Cho x, y, z la cdc so thuTc diTcfng cho xyz = I TiTdng tir CO — i — + >• , _ y+z z+x x+y+2y z+x x+y 2x+y+z + (4) Gia tri nho nha't dat di/dc k h i va chi k h i u = v = w = o A p dung ba't dang thiJc Cosi cd ban, ta c6: x +y + vw iL ie uO nT hi Da iH oc 01 / -L,_L, + uv y +- Khi d6 ta quy ve bai TCf ta c6 minP = - P = x + 2y + z +- v d i dieu k i e n (3) L a i X + y + z = 1, nen 2x + y + z V/ = B l l i toan trd thanh: T i m gia tri Idn nha't cua bieu thiJc P cho diTdi dang (4), 1, ^} 1-z X Khid6tCr(2)suyraPc6dangP TCr (1) va bang cac phep tinh tiTdng tif, ta dtfa P ve dang sau yy dufdc g o i la " B a t d i n g IhiJc Nesbit" (cho ba so) (3) N6 Ik he qua trifc tiep cua bat d i n g thuTc Cosi cd ban Chuy6n dj BDHS6 To&n gii trj Idn nhift Cty TNHH MTV DWH Khang Vi^t gli tr| nh6 nh^t - Phan Huy Kh5i 1.2 Phiftfng phap stf dung trtfc ti^'p bfi't dang thrfc C6si PhiTdng phap thich hcJp vdi nhiTng bai toan tim gia tri Idn nha't \k nho Gid tri Idn nhat dat difdc v^ chi x = y = z = - p^i 2: Cho x, y, z m cac so thuoc khoang (0; 1) va thoa man dieu kien x + y + z = nhat cua ham so' ma co the trifc tie'p ap dung bat dang thiJc Cosi, hoac sau nhffng bien ddi sd cap ddn gian la c6 the diing di/dc bat d^ng thtfc Cosi Tim gi^ tri Ki thuat chu yeu la di/a vao bieu thuTc dau bjli cung nhif cac dieu kien da cho ^^"^^ ^ ~ 1-x chon cac so thich hdp de sau ip dung bat dang thtfc Cosi vdi cAc so a'y + + 1+y iL ie uO nT hi Da iH oc 01 / pat u = - x; v = - y; w = - z 1+x 1 • + + 1+x 1+y 1+z = I 1+x UVW ' f yz Theo ba't dang thtfc Cosi, ta c6: up s/ —^ = o y = z 1+y 1+z yz ( l + y ) ( l + z) Lap luan tiTdng ti/ ta c6: >2 1+y V /g (2) (3) 2) xy (l + x)(l + y) fa w ww Tir (1 + x ) ( l + y)(l + z) > 0, va tCr (5) suy P = xyz < - Da'u bkng (6) xay o =y =z = 1+x dong thdi c6 dau bkng (2) (3) (4) X X = y = Z = - V i le dd suy maxP = - ' ' (2) Bai 3: Cho x, y, z la cac so thiTc dtfdng thoa man xyz = Tim gia tri Idn nha't cua bieu thtfc P = 1 2x^+y^+3 2y^+z^+3 ' 2z^+x^+3' •u ' • -iw Hudng ddn giai 2x' + y^ + = (x' + y ' ) + (x' + 1) + > 2xy + 2x + Do cic ve cua (1) (2) (3) deu 1^ so diTdng, nen nhan tiTng v ' ( l ) (2) (3) v^ cd „ xyz >8— (l + x)(l + y)(l + z) ( l + x)(l + y)(l + z) ' Theo bat d^ng thtfc Cosi, ta cd Da'u b^ng (3), (4) tiTdng iJng xay o x = z; x = y ' Nhtf vay minP = Gia tri nho nha't dat duTdc va chi x = y = z = - (4) ce I +z xz 5^)0 + • om " c •>2 1+x ' Dau bang (2) xay o u = v = w o x = y = z = - + y)(l + z) ok ; u + w > Vmv; TO suy (v + w)(u + w)(u + v) > 8uvw nen tiif (1) ta cd: P > (1) bo Nhir vay ; ^ u + V > 2N/UV ro Da'u bling ( I ) xay o V(l 1+y (1) Ta 1+y ,, UVW Ap dung bat dang thffc Cosi, ta cd v + w > y.+ ^ 1+y 1+z + , „ ^ , • , Luc bieu thffc P cd dang ( l - u ) ( l - v ) ( l - w ) (v + w)(u +w)(u + v) p_ — = HUdng ddn giai ' TCr gia thic't suy u > 0, v > 0, w > va u + v + w = =2 1+z Tim gia tri Idn nhat cua bieu thtfc P = xyz Tir 1-z Hudng ddn giai se cho ta dap so cila bai toan Bai 1: Cho ba so khong am x, y, z thoa man di6u kien 1-y (5) Ttf dd suy 2x^+y^+3 fc-* n.;i, '.i'if, 2xy + 2x + ''^ (1) Dau bkng (1) xay o x = y = (6) TiTdng tif ta cd 1 2y^+z^+3 2yz + 2y + 2z2+x^+3 (2) (3) 2xz + 2z + Dau bkng (2), (3) tifdng lirng xay o y = z = 1; z = x = Cong turng ve (1) (2) (3) va cd P < ^ ^xy + x + yz + y + zx + z + (4) ChuySn 6i BDHSG To^n gia trj Ifln nh51 Da'u bang (4) xay X Cty TNHH MTV DWH Khang Vigt gia trj nhfi nha't - Phan Huy Khii dong thdi co dau bang (1) (2) (3) Liic nky bie'u thurc P c6 dang P = = y = z = X Do xyz = 1, nen ta c6 X = = yz + y +1 xyz + xy + x _ xy • • zx + z + , (5) xy + x +1 xy (xyz)x + xyz + xy ^+ Tir dieu k i e n (2), ta co u ' + = (u + v)(u + w) " u TCr dieu k i e n x + y + z = 1, ta c6 x + yz = x(x + y + z) + yz = (x + y)(x + z) (2) = y = z= - ' ' ww 7i + x ' 7' + y^ Tir gia thiet x + y + z = xyz, la c6 — + — + — = ' , xy yz zx Dat u = - ; V - ; w = x y z Vw + u V U +W + w UV + V W ^ + W +—V y U '* hay P < - (1) (2) • +wu V ^ , , v 2y - i - + — + J- = tanAtanB + tanBtanC + tanCtanA = xy yz zx lanA(tanB + tanC) = - tanBtanC 1-tanBtanC ^^ „ » , n _L ^ = cot(B + C ) o A + B + C = - tanB + tanC ^ P= 'v'v :• = sinA + sinB + sinC • Vr+cot^\l + c o t ' B , > N/I + C O I ' C , s i n A + sinB + sinC ^ A + B + C _ Theo lircJng guic, ta co ^ sm (2) w + V Ta c , => u > 0, v > 0, w > K h i (1) CO dang uv + v w + w u = I w K h i P CO dang: ,,, Vv+w^ V +u W Do X > 0, y > 0, z > va — + — + — = nen dat x = cotA, y = cotB; z = cotC xy yz zx O tanA = (1) W Nhdn xet: X e t each giai bang phiTdng phap liTdng giac hoa sau day: ta c6: Hitting ddn gidi V(w + u"(w + v) Vay maxP = ^ Gia t r i Idn nhat dat diTdc va chi x = y = z = V3 vdi A, B , C e Vi + z^ ^ v U = V = W o B a i 5: Cho x, y, z la cac so ihiTc difdng va thoa man dieu k i e n x + y + z = xyz T i m gia t r i Idn nhat ciia bieu ihtfc P = I •+ V +V + W w T i r d o ta CO maxP = o x • bo x+y+z=l ok z+x=z+y ce o fa D a u bang (2) xay o x = : y = z = - om x+y=x+z y +z=y+x s/ up '"' ro '" /g ', • u = v = w = - ^ c : > x = y = z = V p^(x^y)^(x^z)^(y^z)^(y.x)^(z^x)-.(z.y)^^^^p^^^^^y^^^^^ ' + u + w +V + I + ^^^^ + y ) • (1) A p dung bat dang thiifc Cosi, tijf (1) c6: V Dau bang (2) xay Tifcfng tir, ta cung c6: y + zx = (y + z)(y + x ) ; z + xy = (z + x)(z + y ) NhurvayP " " Tir (1) v a theo baft d a n g thiJc Cosi, ta c6 + yz + yjy + zx + ^J^ + xy Hudngddngiai u 'u+v\u+w B a i 4: Cho x, y, z la cac so ihifc diTdng va thoa man dieu k i e n x + y + z = ' ' ^^^^^"^^^^^ ^ V(v + w ) ( v + u) iL ie uO nT hi Da iH oc 01 / " u ' (3) j • + = (w + u)(w + u), Gia t r i Idn nha't dat di/dc k h i va chi k h i x = y = z = T i m gia t r i Idn nhat cua bieu iMc P = r = (v + u)(v + w ) Wc P c6 dang: P = Thay (5) (6) vac (4) va c6 P < ^ TiT suy maxP = ^ , + , (6) xy + x + , " > sinA + sinB + sinC < — 7r_l ~ 53 ChuySn dg BDHSG Join gii trj I6n nha't va gia t r i nh6 nha't - Phan Huy K h i i Cty TNHH M T V D W H Khang Vi$t' Da'u bang (2) xay x = y = z n c n ket hdp v d i tren suy da'u bang Vay P < I va dau bang x a y r a c i > A = B = C = - < : : > x = y = z = ^/3 Ta thu l a i k e t qua tren B a i 6: Cho X > i v 0, y > 0, z > va X x + y_ X + ]^l4n xet: B a i thi tuyen sinh vao D a i hoc Cao diing khoi D - 2005 diTdi dang bat ^ z+ x yjyz + X dang thuTc sau day: ^zx + y Cho X , y , z la ba so thi/c di/dng va xyz = ChiJng minh P > 3V3 d day P c6 Hitdng dan gidi , y + z = 1, ncn x+y , ^^^^ , 1-z dang nhi/ tren 1-z = , iL ie uO nT hi Da iH oc 01 / ^ Do NhU' vay tir (3) c6 minP = 3\/3 o x = y = z = l ^ y+z yfxy + z ' • + y + z = ; | , / Tim gia tri nho nha't cua bieu thifc P = - ' (3) xay ci> x = y = z = ( v i xyz = 1) R6 rang difdi dang thufc chat la b a i toan tren nhifng "de h(Jn" v i biet tri/dcdapso TUdng tt^, ta CO = Vyz + x , ; = V ( l - y ) ( l - z ) Vzx + y A p d u n g bat dang thufc Cosi, ta c6 • „ a 1-z 1-x Vay P > l(l-x)(l-y)(l-z) V d - y K l - z ) ' V(l-x)(l-y) ' c ok bo 72 zx ww Hitdng ddn giai A p dung ba't dang thtfc Cosi, ta c6 ce „ xy yz zx •••• :h ,.ht lAjliii' fa '^y V l + z^ + x ^ hay P > xy 1-x^ 1-y^ •+ Vy/ 7- + -7 •> ' z +x +y X ,i 1-z^ A p d u n g b a t d ^ n g thuTc Cosi, ta c6 ; = 2x^ + (1 - x ' ) + (1 - x ^ ) > ^ \ ^ ( \ - \ ^ f 8>27.2x'(l-x')^^ ^>x^(l-x^)^ (2) 2 V i O < x < l n e n t i r ( ) suy x ( l - x ' ' ) < — ^ 3V3 ^^^'^TT^- • ^a.v.:a: 3V3 o -!—< xd-x^) A.,"^^^^^ Da'u bang (3) xSy 2x^ = - w , ^ x/i ++yy^ + z ^ /g ,^iu[...]... m i c n | xac d m h cua no 29 Cty TMHH MTV D W H Khang Vi$t Chuyfin dg BDHSG loan g\i trj Idn nhat va gia trj nh6 nhift - Phan Huy KhSl HUdng ddn gidi Tir do suy ra P > 8, vay minP = 8 Xem Idi giai trong bai loan 1, muc §1, chiTdng 1 cuo'n sach n^y Cach giai nay sai d ch6 la mdi difa vao phan 1 cua dinh nghTa gid tri nho nhat Ta xem phan 2 cua dinh nghTa c6 thoa man hay khong? De y r i n g da'u giai... B ia tap cua D, trong do A e B "^'^2 (1) 'h D | = {(x; y): x > 0 ; y > 0 ; 4 < x + y < 6 } , D2= {(x;y):x>0;y>0;x + y ^ ro net... gia t r i Idn - 0 Ta ci) I ' d ) - 1 - I- HUdiig dan giai I ' -1 — r Do x 7^ 0 , y ;t 0 n c n xy(x + y) = x ' - xy + y" • y : • Cty TNHH MTV DWH Khang Vi$t Chuy6n dg BDHSG Jo&n gii tr| I6n nhflt 1 1 1 O 1 1 - + - = — + — X g\i trj nh6 nha't - Phan Huy Khi\ 2y y ' xy (1) ^ • i D a t X - - ; Y = - k h i do (1) c6 dang: X + Y = X y - XY + YI f'(y)- (2) V d i dieu k i e n (2) t h i A = ( X + Y ) l Vl2y>...Chuyfin dg BDHSG Join g'lA tr| Ifln nha't vt g\i trj nh6 nha't - Phan Huy KhAi Cty TNHH MTV DVVH Khang Vi^t Da'u bkng irong (7) x a y ra khi va chi khi (diiu bang xay ra ehi lai t = 0), n c n r ' ( t ) la ham nghich bien trcn |(); - J b... 1^ - + k X y — + - tri Idn nhat v^ nho nha't cua ham so c6 the se dan den sai lam y) Taco x + - > 2 ; y + - > 2 v a - + ^ > 2 X y y X 31 Cty TNHH MTV D W H Khang Vigt Chuy6n dg BDHSG ToAn gia tr| Mn nhift vji glii tr| nh6 nha't - Phan Huy KhJi B Cdc tinh chat cua gid tri l^n nhat vd nho nha't cua ham so min f(x) = min min f(x); min f(x)|> Gia sur ton tai max f(x); max f(x); min f(x); min f(x) xeA Khi... ( 7 - x) + 2 - V(x + 3 ) ( 7 - x ) ( x + 2 ) ( 5 - x ) T a c o : f ' ( l ) = 2t + 3 + Vl-2t r(t) = 2 - (2) g\& trj nh6 nhS't - Phan Huy Khii 2 D o y > 0, nen CO y> V2 , V X e Cty TNHH MTV DWH Khang Viet [-2; 5] V i the ta C O ( - x ' + 3x + 10)(2x - 4 ) ' - ( - x ' + 4x + 21)(2x - 3 ) ' = -51x^... DC-thay ming(x) = - ^ ; min h(x) = 16.^ = 8 xeD • xeD Dau b a n g trong (2) x a y ra k h i va chi khi Ion tai x„ e D sao cho sin" 2x 37 Cty TNHH M T V DVVH Khang Vjgt^ Chuyfin 66 BDHSG Toan gia t r i Idn nha't va gia trj nh6 nha't - Phan Huy K h i i minf(x) = f(X(,);maxg(x) = g(x,)), xeD ; xeD •', 2 Neu them vao gia thie't f(x) > 0 V x e D Khi do vdi moi n nguyen dtfdng, , ta co: max f(x) = 2n/max f^"(x);