Giá trị lớn nhất, giá trị nhỏ nhất - Chuyên đề bồi dưỡng học sinh giỏi: Phần 1

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Phần 1 tài liệu Chuyên đề bồi dưỡng học sinh giỏi - Giá trị lớn nhất, giá trị nhỏ nhất cung cấp cho người đọc các kiến thức mở đầu về giá trị lớn nhất và nhỏ nhất của hàm số, phương pháp sử dụng bất đẳng thức để tìm giá trị lớn nhất và nhỏ nhất của hàm số. Mời các bạn cùng tham khảo.

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0+150^7^ P.GS - IS PHAN HUY KHiH Chuqen de / 01 oc iH BOI DUONG HOC SINH Da hi nT uO ie Gia tri I0n nliiK iL Ta s/ Gia tri nli6 nlid up ro /g om .c ^Danh clio hoc sinh Idp ok bo ce >BfensoantheonOidun^va fa w. c^utrucd^tliicuaBOGDfiflT ww • ^ V I E N Tl'NHBiNHT OK] NHA XUAT BAN DAi HQC QUOC GiA HA NQI Ha NQI
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 IJCU N6I i>Au 16 Hang Chuoi - Hai Ba Trang - Ha Npi Bai todn tim gid tri U'fn nhd't, nhd nhd't ciia ham so noi rieng vd hat dang thiic ndi Dien t h o a i : Bien t a p - Che ban: (04) 39714896 chung Id mot trong nhifng chii de quan trong vd hu'p dSn tnmg chutfng trinh gidng day vd Hanh chinh: (04) 39714899; Tong bien tap: (04) 39714897 hoc tap In) mon Todn d nhd trudng phd thong. Trong cdc de thi mon Todn ciia cdc ki thi Fax: (04)39714899 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 trong Chiu trdch nhiem xuat ban nhiing cdu kho nhd't ciia de thi. , , ., , / Vdi li do do cdc cud'n sdch chuyen khdo ve chii de nay ludn luon thu hut su chii y vd 01 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, oc gid tri nho nhd't" nay, chiing toi se cung cap cho ban doc nhvtng cdch gidi thong dung iH Bien tap vd sita bdi: H A I NHtf nhd't doi vdi nhiing hdi todn tim gid tri Idn nhd't vd nhd nhd't ciia hdm so.cdng nhu hiet Da Che ban: Cong ty K H A N G V I E T cdch dp dung hdi todn nay de gidi nhieu hdi todn lien quan den no. hi Noi dung ciia cud'n .sdch dUOc trinh hdy trong chUcfng. nT Trinh bay bia : C o n g ty K H A N G V I E T Chiiong 1 v(H tieu de " Vdi bdi todn md ddu ve gid tri l^n nhd't va nhd nhd't cua ham uO Chiu trdch nhi^m ngi dung vd ban quyen so" se gidi thi^u vdi ban doc bdi todn tim gid tri Idn nhd't, nhd nhd't ciia hdm .sd'thong ie Cong ty TNHH MTV DjCH Vy VAN HOA KHANG VI^T qua vi^c trinh hdy tinh da dang ciia cdc phUcfng phdp gidi hdi todn ndy. Bdng cdch diem iL lai nhiing .sU cd m$t ciia cdc hdi thi ve chii de ndy cd mdt trong cdc ki thi tuyen .nnh Dai Ta Tong phdt hdnh: hoc - Cao dang cdc ndm tic 2002 den 2011, cdc ban se thd'y duac sU can thie't cua vi$c s/ phdi trang hi cho minh nhvtng kien thiic de gidi quyet cdc hdi todn d'y. Cud'i chUtfng 1 Id up cit sd li thuyet ciia hdi todn tim gid tri Idn nhd't vd nhd nhd't ciia hdm so. Phun nay giup ro C6NG T Y TNHH MTV cdc ban nhiing kien thiic chud'n hi can hiet di' doc tiep cdc chUifng sau ciia cud'n sdch. /g Sm ajP D ! C H vy V A N H 6 A K H A N G V I | T om Cdc phUcfng phdp ca ban vd thong dung nhd't de gidi bdi todn tim gid tri Idn nhd't vd nhd nhdt ciia hdm sd'duac trinh hdy tit chUOng 2 den chuang 6. ., • Chitang 2: Phi/mg phdp h&t ding thuCc tim gid tri l^n nhdt vd nho nhdt cua ham sd. .c /^Dia chJ: 71 Dinh T i § n Hoang - P D a Kao - Q.1 - TP.HCM ~ ^ 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 ok Fax: 08. 3911 0880 so'. bo Email: l
Cty TWHH MTV D W H Khang Vi^ ChMng 7 danh de trinh hay vi$c ling dung ciia hai todn tint gid tri U'fn nhd't, nhd nhat trong vi^c hi$n ludn phu
Chuyen BDHSG Toan gia tr| Idn nha't g'A tr| nh6 nhat - Phan Huy KhJi Cty TNHH MTV DWH Khang Vi§t + 1 + 1 > 3z. Tir do va diTa vao gia thie't x + y + z = 3 suy ra: + y ' + z^ > 3. (2) P- (I) x-y/x^ + 8 y z y-y/y^ + 8 z x z^z^ + 8 x y D a u bang trong (2) xay ra o X = y = z = 1. TO'(1) va theo bat dang thiJc Svac-xd, ta c6: T i i r ( l ) , ( 2 ) s u y r a P > 1. (3) (x + y + z ) ' D a u bang trong (3) xay ra dong thdi c6 dau bang trong (1), (2) ' (2) x^jy} + 8yz + y ^ y ^ + 8zx + z-y/z^ + 8xy , . -. X = y = Z = l . Z^.,-, A p dung bat dang thiJc Bunhiacopski, ta c6: / V a y m i n P = 1 x = y = z = l . 01 B a i 14, Cho x, y, z la cac so' thifc di/dng. T i m gia tri nho nhat ciaa bieu thtfc X/X.N/X^X^ + 8 y z + ^/y.^/y^/y^ +8zx + N/Z.VZI/Z^ + 8 x y oc r. y^ z^ iH 2 2 2 2 2 2 (x + y + z) x ( x ' + 8 y z ) + y ( y ^ + 8 z x ) + z ( z ^ + 8 x y ) Da y +yz + z z +ZX + X x +xy + y hi Hiidngddngiai ;;':..(•'•...!; '-„4/.!' j,, = (x + y + z ) ^ x " ' + y ^ + z ' ' + 2 4 x y z j . * ' ' (3) nT V i e t l a i P dtfdi dang: 4 4 4 . . A p dung ba't dang thiJc Cosi, ta c6: • • , ^1 uO P= + y_ + I (1) (X + y + z)^ = x^ + y ' + z' + 3(x + y + /)(xy + y/ + /x) - 3xyz ie x ^ ^ y ^ + y z + z^j iL y^^z^+zx + x ^ j z^(x^ + xy + y ^ j > x ' + y-'+ z ' + 277xyz.\/xVz^ - 3xyz Ta A p dung bat dang thufc Svac-xd, ta c6: * hay (X + y + z ) ' > x ' + + z' + 24xyz. , . , ,, . (4) s/ fx^+y^+z^f P> i L up Thay (3), (4) vao (2) va c6: P > ^^ ^ = 1. (5) x^(y^ + y z + z^) + y^ (z^ + z x + x ^ j + z^(x^ + x y + y^) (x + y + z)^ ro Is!,. Jr. ; De thay da'u bang trong (5) xay r a o x = y = z = 1. /g , x^+y^+z^+2(xy+yV+zV) hayP> ^ ^ . . (2) Nhqn xet: Ta c6 bai toan IMng tU" sau: om 2^x y + y z + z X j + ( x y ) ( y z ) + (xy)(zx) + (yz)(zx) .c Cho X > 0, y > 0, z > 0 va X + y + z = 1. T i m gia tri nho nhat cua bieu thtfc I ok Theo bat d i n g thtfc Cosi, ta c6: i P = -+ • bo x' + y' + z ' > x y + y V + z V (3) x''+8yz y-^+8zx z^+8xy ce x^y^ + y^z^ + z^x^ > (xy)(yz) + (xy)(zx) + (yz)(zx) (4) z^ Ta giai nhu" sau: P = — + fa (*) z" + z'' + z > 3z^ X' + 8 x y z y +8xyz /: + 8 x y z w. 3(x^y^+y^z^+z^x^] 4^- ww Tir (2), (3), (4) suy ra: P > — — ( hay P > 1. (5) A p dung ba't dang thuTc Svac-xd, ta c6: P > (x + y + z) (**) 3(x^y^+yV+z\^j x' + y + z" + 24xyz De thay dau b^ng trong (5) xay ra o X = y = z > 0. Theo bai tren ta c6: (x + y + z ) ' > x V y ' + z ' + 24xyz. ^ Vay min P = l < i > x = y = z > 0 . (x + y + z)^ 1= Tilf (**), (***) suy ra: P > " \y P > 1. Bai 15. Cho x, y, z la cac so thifc diftftig. T i m gia t r i nho nhat ciaa bieu thuTc (x + y + z)- x +y + z +8yz >/y^ + 8zx + ^xy Vay min P = 1 o X = y = z = ^ . Hiidng ddn giai ^ a i 16. Gia siif x, y, z la ba canh cua mot tam giac c6 chu v i bang 12. V i e t l a i P difdi dang sau: 1 ^ 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 ra x = V 4 - x ^ o x = V2 . - 2 < F((p) < 2V2 V - | < ( p < ^ , Vay maxy = y/l C:>x = yl2 . 71 Cdch 2: (PhiTOng phap chieu bie'n thien ham so) F((p) = 2%/^ cos = 10 9 = — X = N/2 , ; • 4 Xet ham so f(x) = x + V4 - x^ vdi - 2 < x < 2 71 V2 71 37t 71 F((p) = - 2 cos CP-- (p — = o (p = — x = - 2 . X v4-x^-x Ta CO l"'(x) = 1 I I S i 2 ^ 4 4 ' 2 / Vay max Hx)= max F((p) = 2>/2; 01 /4-x^ \/4-x' '5 -2 0 khi 0 < X < yfl va 4 - 2x' < 0 khi N/2 < x < 2, nen ta c6 bang hi Cdch 4: (Phu'dng phtip mien gia trj ham so) bien thien sau: nT 0 ^/2 ^^^'..2. Gia sur m hi mot gia trj tiiy y cua ham so \'(x) = x + \l4-x' vdi - 2 < x < 2. uO Khi do phu'dng trinh an x sau day x + \J4-x~ =m (1) c6 nghiem. ie iL Ro rang (1) o \ / 4 - x - = m - x. (2) Ta B^i loan ltd thanh: Tim m de (2) c6 nghiem. ^" ' s/ Tirdosuyra max l"(x) = I(N/2) = 2N/2 ; (2) CO nghiem khi vii chi khi difclng cong y = SJA-X^ va diTcJng thang y = m - x up •"''^ " min l(x) = min{l'(-2);r(2)) = min(-2;2) = - 2 i > : cat nhau. n . . . : , . v - , ! - . - x , ro - 2 < ,\ 2 /g De lha'y y = m - x o x + y = m, con y = ^ 4 - x " c^< ^ ~^ Nhgn xet: Ten goi cua phifdng phap hoaii toan phan linh di'ing qua each giiii vifa om I y>0 trinh bay cJ' Iren. .c Vay ta can tim m de du'c'Ing thang x + y = m va nifa du^clng Iron x^ + y" = 4 ok Cat7i J; (PhifcJng phap UMng giiic hoa) i (phan nam phia tren Iriic hoiinh cat nhau). bo Xet ham so t(\) = \+ ^ - x ' \(U - 2 < x < 2 Dc tha'y dieii nay xay ra ce Do - 2 < X < 2, nen dat x = 2sin(p vdi - - ^ < cp < ^ . khi va chi khi du'cVng fa TCr do ta quy ve xet ham so thring X + y = m nam w. giiJa hai du'clng x + y = - 2 ww F((p) = 2sin(p + >/4(l - sin" (p) 2sin(p + 7 4 c o s ' cp = 2sin(p + 2|eos(p va X + y = 2 V2 , ti'fc la = 2sin(p + 2cos(p (do khi - - ^ < cp < ^ ihi coscp > 0) khi vii chi khi - 2 < m < 2V2 .(3) = 2N/2COS((P--). Tir(3)suyra ,^ n n 37t 71 rt ;,i;.f- max r(x) = 2%/2; Do — < ( p < - => - • — < ( p - - < - . -2
Cty TNHH MTV DWH Khang Viet Chuygn dg BDHSG Toan gia trj Icin nha't gia tri nh6 nhat - Phan Huy Khtii Nhdn xet: Cach giai tren diTa vao each tim gia tri cua ham so day c6 ket hdp (x-y)(l-xy) (X + y)(l + xy) T i i r ( l ) ( 2 ) suy ra (3) i2 • them phifdng phap suT dung do thi va hinh hoc), vi the ta c6 the noi rang da (l + x ) ' ( l + y)^ [(x + y) + (l + xy)] sur dung phi/dng phap mien gia tri ham so' de giai bai toan tim gia trj \6n nhat Mat khac d i lha'y [(x + y) + (1 + xy)]^ > 4(x + y)(l + x y ) . (4) va nho nhat noi tren. Dau bang trong (4) xay ra « x + y = 1 + xy. Binh ludn: Vdi bai loan 1, la da su* dung bon phu'cfng phap khac nhau de giai (x-y)(l-xy) bai loan tim gia trj Idn nhat va nho nhat cua ham so. M o i phU'dng phap deu Tir(3) & (4) di den 0, y > 0. Tim gia tri Idn nhal va nho nhat cua bieu thiJc P = x = l ; y = 0, khi d6P = - oc xy = 0 [xy = () (x-y)(l-xy) •: ^ •• ' • • ' iH Lai CO P 4 x + y = l + xy [x + y = l x = 0;y = l , k h i d6P = - i (l + x ) ^ l + y)2 Da • hi (De thi tuyen sink Dai hoc, Cao ddn^ khoi D ) nT ( . i M I HuAng ddn giai Tom lai maxP = - < = > x = I ; y = 0; minP = — x = 0; y = 1. 4 4 uO CacA 7; (Phufdng phap ba'l dang ihiJc) CacA J ; (Phu'dng phap ba'l dang Ihtfc) • ' ie lha'y P c6 the vie't lai dudi dang sau day iL AB . ta co: ( x - y ) ( l - x y ) ^ ( x - y . 1 - xy) : X y _ X 1 y 1 2/1 , . , \ n j_ v^2/-l . v"!^ P = (l + x)"(l + y ) ' A4(l + x ) ' ( l + y) Ta (1 + x)^ (1 + y)^ (1 + x)^ 4 (1 + y)^ ^ 4 s/ ( x - y + l - x y ) 2 ^ (l + x ) 2 ( l - y ) ' _ 1 4y 0) 1-- up 4 x - ( l + x)' 1 - i _ (x-ir , y Lai eo 4 (1) 4(l + x ) ^ l + y)- 4(l + x)2(l + y)^ 4 (i + y r 2 4 ~ 4 ro 4(1+ x)' (1 + y) 4(l + x)2 {\ yf /g Tird6suyraP^^^-y^/^-^^[4 Vx.O;y^O Do y > 0, nen lij" (1) suy ra P < - , V X> 0, y > 0. P = - x = 1; y = 0. (l + x)^(l + y ) ' 4 om .c TiTOng liTlai c6 ^' '< i5«»f^« 'flfffetv''}l>_'n^frflJ j s l i i i . f U ' Mat khac P = - ' ^ ^ . VaymaxP= - x = l ; y = 0. 'f' ok 4 x=l 4 1 1 X ^ (y-1)' bo P= (2) Do vai iro binh dang giffa x va y, nen la co (1 + x) (1 + y)^ " 4 4 ce (y-x)(l-yx) ^ 1 . p _ , U - y K l - x y ) > _ ivx>(),y>0. fa Do X > 0, nen liJT (2) suy ra P > - - V x > 0; y > 0. P = - - o x = 0; y = 1. (l + y ) 2 ( l + x)' 4 (l + x)^I + y)^ w. 1 ww Tom lai max P = — < = > x = l ; y = 0; minP = x = ();y = 1. Mat khac P - - - o x = ( ) ; y = 1. V a y n e n P = o x = 0 ; y = 1. 4 4 Nhdn xet: Cung suT dung phiTcMg phap ba't dang thiifc, nhu-ng ta co 3 each giai khi Cach 2: (PhiTdng phap bat dang ihu-c) (x-y)(l-xy) X - y 1 - xy nhau bai loan tren. Ta c6: (1) , Cach 4: (Phifcfng phap lifting giac hoa) '1 i (l + x ) ^ l + y)2 (l + x)^(l + y)^ Do X > 0; y > 0, ncn hien nhien la c6 Ta co: P = (xem each 1). (1 + x)^ (1 + y)^ x - y | | l - x y | < ( x + y ) ( l + xy) (2). .i Do X > 0; y > 0, nen dat x = tan^a, y = tan^(3, ( ) < a < - ; 0 < p < - - Dafu bang trong (2) xay ra o xy = 0.
ChuySn BDHSG To^n gi^ tr| Idn nha't vA gia trj nh6 nha't - Phan Huy Khii Cty TNHH MTV DWH Khang Vi^t K h i do P = , ^ ^ " ' f , = lan^acosV - tan^pcos^p 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) = -li—Ili^, k h i d o phiTcIng t r i n h t^+2l4-3 (1 + tan^ar (1 + lan^ p)^ • . sau d a y ( a n t ) :' , = sinWos^a-sin'Pcos'P=-sin'2a- — sin'2p. (1) -•^'^ "^'^^ = m ( 2 ) C O n g h i c m . D e tha'y v i t ' + 2 l + 3 > 0 V l , t^+2t + 3 TOr(l)suyra - - < P < - Va, pG|();-). 4 4 2 nen (2) • » 2 1 ' + 12l = m ( t ' + 2t + 3 ) / n (m - 2 ) t ' + 2 ( m - 6)1 + 3 m = 0 . : (3) 01 sin 2a = 1 a =— X - 1 L a i l h a y P = ^< 4 i * N e u m = 2 , k h i d o 2 ( m - 6 ) ^ 0, nen ( 3 ) c 6 n g h i c m . V a y m = 2 la m o t g i a oc sin2p = ()'^ y = ()' p =o t r i c u a h a m so r(t). iH a =0 * N e u m^l, k h i d o (3) c 6 n g h i e m k h i va c h i k h i A ' > 0 Da sin 2 a = 0 x-O P= — « c:> m ' + 3 m - 18 < 0 o - 6 < m < 3. hi 4 sin2p=I y = i V i i y ( 3 ) CO n g h i c m k h i v a e h i k h i - 6 < m < 3. ,/ / (4) nT 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 " ( 4 ) suy r a ,r ' uO V a y m a x P = ^ < z > x = l ; y = (); m i n P = - - ^ x = 0 ; y = 1 . i*> m a x P = max r(t) = 3 v a m i n P = m i n r(t) = - 6 . , 1 ! ie y*() ItR y*l) lelR iL liinh luan: V d i biii loan t r c n la c 6 4 each g i a i khac nhaii, Irong d o c 6 3 each Ket hdp P = 2 k h i y = 0, l a d i d e n k c l l u a n : ^ Ta 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 V d i d i e u k i e n x ' + y^ = 1 i h i m a x P = 3, m i n P = - 6 . s/ 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 Cat7i 2 ; (PhU'dng p h a p m i e n g i i i t r i h a m so) ' up n h o nha't eiia 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]. ro IJai t o a n 3 : G i a silf X, y la hai so ihifc sao c h o X " + y^ = 1. , . 2sin" a + 12sintteosa 1-cos2a+ 6sin2a ..^ /g KhidoP= — = ^ - 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 = +6xy)^ 1 + 2sinacosa +2cos'a sm2a + cos2a + 2 . , om 1 + 2xy + 2y" G o i m l a g i a t r i t u y y c i i a P. , , .c ' (' , (De thi tuyen sink Dai hoc. Coo ddn)> khoi B) 1-cos2a + 6sin2a ok K h i d o phu'dng t r i n h sau d a y ( a n a ) —=m (l) Hitdng ddn gidi sin 2 a + cos 2 a + 2 bo ':dch 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) ' .i . < {i C O n g h i t M i i . D o |sin2a + c o s 2 a | < V2 , V a e |(), 2TC| ce 2(x^+6xy) •' fa Do X' + y " = 1, n e n ta e o : P = => s i n 2 a + c o s 2 a + 2 > 0 V a 6 |(); 27i|. ; , (1) x^ + 2 x y + 3y^ ' w. Tir do (2)o 1 - cos2a + 6sin2a = m(sin2a + cos2a + 2) ww X e t h a i k h a n a n g sau: o ( 6 - m ) s i n 2 a - (1 + m ) c o s 2 a = 2 m - 1. (3) 1. N e u y = 0 ( k h i d o x = 1). L u c n a y P = 2. (3) CO n g h i e m 0 ( 6 - m ) " + (1 + m ) ' > ( 2 m - 1)" -> •., X c ^ 2 m - - 3 m - 9 < ( ) o - 6 < m < 3 . (4) + 6 T i r d o suy ra m a x P = 3, m i n P = - 6 k h i x - + y~ = 1. 2t- + i2t , ; X . 2. N c u y ^ 0. K h i d o P = Cdch 3: (PhiTctng p h a p c h i e u b i e n ihiC-n h a m s o ) j " / \ — , (1 day I = — va t e X -2V3 t"+2t +3 y TacoP^ 2(x^.6xy)^ (xemcachl). X + 2xy + 3y" * N e u y = 0, l h i P = 2. ;
Cty TNHH MTV D W H Khang Vigt Chuy§n dg BDHSG Tpan gi^ tr| Ifln nhat gia tri nh6 nhat - Phan Huy KhSi * Ncu y ^ 0, thi P = ^^^^ vdi t = - 6 Ap dvng (2) vdi a = ^ ; b = f. Kb. do a > 0, b > 0 va ab = - . 1 (do x . y), r +21 + 3 y (4) Dat f(t) = t G R thi f'(t) - -81^ + 121 + 36 , , 2t^-3t-9 4 nen ta c6: I_+-L->-^ t^+2l + 3 ( l ^ + 2 t + 3)^ ( t ^ + 2 t + 3)^ ' 1+^ l ^ ' ^ 1+ V z Ta CO bang bie'n thien sau: 2 ii iH: z _ X I -00 3 +O0 / ^ =z 01 Dau bang trong (4) xay ra y x=y oc I'd) 0 + 0 - iH 1(0 2 2 y y Da 1 (5) 2 Tif (4) ta CO P> + hi X Tir do suy ra max f(l) = 3 va min t'(t) - - 6 . 2+3^ 1+ nT teE leM U'W 'n'.l-.U:^:'' 'f.f • ' X Vy uO Vay maxP = 3, minP = - 6 khi x^ + y ' = 1. 7y^ = z ie Binh luqn: Tinh da dang cua cac phU'dng phap giai bai loan lim gia tri Idn nhat Dau bang trong (5)xayra x =y iL A i .... i ;l I va nho nhat cung the hicn ro qua thi du nay. Ta Bai toan 4: Cho x > y, x > z va x, y, z e [ 1 ; 4].Tim gia tri nho nhat ciia bieu Datt = E . D o x > y v a x . y e l l ; 4 ] n e n s u y r a l < ^ < 4 = ^ l < t < 2 . K h i d 6 s/ x y z thiJc: P = up 2x + 3y y+z z+X \ t 2 ?>—!— +—-hay P > - ^ + 7~T- ro (De thi tuyen sink Dai hoc Cao ddiifi khoi A - 2011) 3 1+t 21^+3 l + l 2+ /g Hii(fng dan gidi om LtJi giiii cua bai toan nay la su" kel hctp khco Ico cua hai phifcfng phap bat Xet ham so' f(t) = + - 1 ^ vdi 1 < t < 2. .c dang ihiJc va chieu bien thien ham so nhif sau: 21+3 1+t ok 1 1 1 61 2 (31 -61^) + ( 3 1 ^ - 4 1 ^ - 9 bo Viet lai bieu Ihtfc P diTdi dang: P = •+ + (1) Taco: f'(t) = ^ ^ ^ - ^ = (21^+3)^(1 + 1)^ ^ ce 2 + 3^ 1+^ 1 + '^ y VI t > 1 ^ f (t) < 0 V t e [ 1 ; 2 ] . TO do c6 bang bien thien sau: fa • ' ' ' ' • X w. Tru"(1c hel ta chiJng minh bat dang thiJc sau: * " ' *' • ^ I 1 1 1 2 —•— ww Neu a > 0, b > 0 va ab > 1, Ihi la c6: + > 1 + a 1 + b 1 + x/ab (2) f'(t) 1 Dau bang trong (2) xay ra khi va chi khi a = b hoac ab = 1. f(t) i Thatvay(2) Vay min f (t) = f (2) = | 1 . TO do suy ra P > ^ , l 0 o 7^) > 1+a l + >/abJ 1^1 + b 1 + Vab (l + aKl + x/ab) (l + b)(i + (V^-^/b)^^/^-l) >0. (3) (l + a)(l + b)(l + >/ab) Do a > 0, b > 0, ab > 1, vay (3) dung suy ra (2) dung.
gii KhAi Chuygn dg BDHSG Toan trj I6n nha't va glA tr| nh6 nha't - Phan Huy - Phiftlng phap mien gia tri ham so. 'i--^ j^ifif'^f^'M^^'•''•>::^r^' Do X, y, z e [ 1; 4] ncn P = — x = 4, y = l , z = 2. - Phi/dng phap lU'dng giac hoa. ..c.^i '.-rl- 33 - Phi/ctng phap hinh hoc hoa. ,;:^^^iry, h^> rti - . 34 Nhu" the minP = x = 4;y = l;z = 2. Cac ban cung da tha'y dtfdc chiing ta c6 the c6 nhieu phU'dng phap khac 33 ; ~" nhau de giai cung mot bai toan tim gia tri Idn nhat va nho nhat cua ham Bai toan 5: Cho bon so ihifc a, b, c, d thoa man dieu kien a^ + b" = c' + d"^ = 5. Tim gia trj Idn nha'l cua bieu thiJc P = > y 5 - a - 2 b + V 5 - c - 2 d +N/5 - a c - bd . §2. N H I N LAI C A C BAI T O A N V E GIA TRj L 6 N N H A T V A / Hii(fng dan gidi 01 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 oc Ldi giai hay nhat va dac sSc nhat cho bai loan nay la phu^dng phap su* dung SINH V A O D A I H Q C , C A O D A N G iH hinh hoc sau day: Cac bai toan tim gia trj Idn nhat va nho nha't cua ham so thu'dng xuyen xua't Da Ta thay cac diem M(a; b), N(c; d) va Q( 1; 2) trong do a, b, c, d la cac so thifc hien trong cac ki thi tuyen sinh vao Dai hoc, Cao dang nhiTng nam gan day. hi thoa man dieu kien dau bai deu nam tren difdng Iron c6 tam tai go'c toa do Trong muc nay chung toi xin gidi thieu cac bai toan ay kem theo nhffng binh nT va ban kinh bhng v 5 . ' f . i"i: V ^^ luan can thiet. , ,/ • ^ v uO Viet lai bieu iMc P dxidi dang sau: f ? x^vlfi^m- ie /(a-l)2+(b-2)^ Bai 1: (De thi tuyen sinh Dai hoc Cao dunf- khoi A-2011) ka-cf+ih-df iL P= Cho X , y, z la cac so thiTc sao cho x > y, x > z va x, y, z e [1; 4). Tim gia trj Ta ^ t X y z nho nha't cua bieu thuTc: P = + + . 7 "• :: - s/ (MQ + NQ + MN) = up 2x + 3y y+ z z + x • l; ' 1^)'?' d day CMNQ la chu vi cua tam giac MNQ. HUdng ddn gidi ro Ta sur dung ke't qua quen bict trong hinh /g Xem Idi giai trong bai toan 4, muc §1, chu'dng 1 cuon sach nay. hoc phiing sau day: Trong cac tam giac om Binh luan: .c npi tiep trong mot di/dng tron ban kinh 1. Mau chot de giai bai 1 la d cho bang each su" dung mot ba't dang thiJc ok R cho trU'dc, thi tam giac deu la tam bo giac CO chu vi Idn nhat. Mat khac tam phu, de diTa ve danh gia P > — ^ — + — y . (1) ce giac deu noi tiep trong du'dng tron c6 fa ban kinh R c6 canh b^ng R ^/3 . De'n day b^ng each difa vao an phu t = ^ -• vdi t G 11; 2] ta quy v^ danh w. y ww VivayCMNQ< 3 N / l 5 . T i r ( l ) s u y r a = -N/30. gii P > + _ L . f(t). P < ^ ^ '^H*>v^^-; • 3^30 v2 2 2t2+3 1 + t Do d6 maxP = R6 rang tiep theo ta nghT ngay den se sur dung phiTdng phap chieu bien . Qua 5 bai toan tren, chung toi da gidi thieu vdi c&c ban cac phiTdng phap? thien ham so de tim min f(t) vdi 1 < t < 2. chinh de giai bai toan tim gia tri Idn nhat va nho nhat cua ham so". Tir do ke't hdp hai qua trinh tren ta se di den IcJi giai cho bai toan. Van d6 - PhiTdng phap bat d^ng thtfc. la d cho viec phat hien ra (1) khong phai la dieu de dang. - PhiTcfng phap chieu bie'n Ihien ham so.
Chuy6n dg BDHSG Toan gii trj I6n nha't va gi^ tri nh6 nha't - Phan Huy Kh5i Cty TNHH MTV DWH Khang Vigt 2. Thay cho vice suT dung mot bat dang thtfe phu, ta co each lam sau day c6 Bai 2: (De thi tuyen sink Dai hoc Cao ddrtfi khoi 8-2011) ve "tiT nhien " hcJn mot chiit. Cho a, b la hai so thiTc dUdng thoa man dieu kien: Coi P nhiT la mot ham so'eua z, xet ham so'an z. 2(a^ + b^) + ab = (a + b)(ab + 2) ' P = P(z)= — + - ^ + - ^ vdiz G [l;x]. b^^ Tim gia tri nho nha't cua bieu thtfc P = 4 + 9 b^ a^ '>;< 2x + 3 y y + z z + x + ) Khido P'(z) = () L + Hiidng dan giai (y + z)^ {z +X xf _ x(y+ {y +z)^-y(x zfiz + +xfz)^ _ f a h\ a b\ a b^a b / (x-y)(z^ - x y ) Difa P ve dang sau: P = 4 Vb— + &) — -3 + 9 b + a— - 2 01 — (1), h— b a oc {y + zf(z + xf b a\ a b a b^ iH a b * N e u x ^ y , lhiP(z)= ^ + ^ ^ + - ^ = ^- V z e [ l ; x ] = 4 — + — + 9 b ^ a j - 1 2 —b + —a 18. (1) • Da .b a j 5y y + z z + y 5 hi Viet lai gia thie't diTdi dang sau: 2 r a b ' +1 = (a + b) 1 + - - . .. i nT * Ne'u X > y (chii y la x > y, nen khi x y thi x > y) thi x - y > 0 nen b aJ ab uO ii^.'"' P'(z) = 0z^-xy = 0z= ^xy • aiaj',;;:.;J.J:-:^'^ 2 '2,\/^ Theo baft dang thtfc Cosi, ta c6 I + — > - p = . .Hi!K ie Ta CO bang bien thien sau (suy ra tif (1)) ? , ! ' !7 ab vab iL z 1 2 1 ^a b " + l > ^ ( a + b) = 2V2 1 Ta P'(z) 0 + Thay (3) vao (2) va c6: 2 vu + ay — Vab Vb ^ Va .(4) i r — b a s/ P(z) up ^ - ^ ^ - — ^ ro Vay vdi mpi z e [1; x], ta c6: P(z) > /g om Khi do tir(4) ta c6 2(t^ - 2) + 1 > 2 > ^ t hay 2t' - 2 V2 t - 3 > 0 p(7^)= X y =>(>/2t+l)(N/2t-3)>0. (5) .c 2x + 3y " y + ^ " 7^ + x Do t > 2 => t + 1 > 0, nen tiif (5) suy ra ' ok p ( z ) > ^ - ^ ^ Jy ^ bo V2t-3>0=>t>^=>i + ^=t^-2>l. (6) 2x + 3y + slx+yfy ce ^/2 b a 2 P = P(z)> — -1— + —2 fa (2) Bai toan quy ve: •>^'-*v-'.^*'.'^^ 5 w. Tim gia tri nho nha't cua ham so f(t) = 4t^ + 9t^ - 12t - 18 vdi t > - . ww Da'u bang trong (2) xay ra o z = ^xy . Ta c6: f'(t) = 12t^ + 18t - 1 2 = 6(2t^ + 3t - 2) va c6 bang xet dau sau: 6 34 1 t -2 2 Den day ta lie'p tuc giai nhiT phan sau cua bai 4, muc §1 v6i lifu y rkng do - >— 5 33 f'(t) + 0 2 0 + 1 + i f(t) 34 23 23 Vay minf(t) = f 2. = - —4 . t t f c m t a c d P > - — (7) '4 nen minP = — 4 33 Ro rang viec phat hien ra (2) theo cdch giai nay "tif nhien" hdn trong each giai cua bai 4, mat du no phuTc tap hcfn ve mat tinh toan! 15
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 • • , V t G [0; - 1 . _a _ J_ Tcrdc) r(t)>r b a ,b ~ 2 Do f ^ 11 _ 2V3 >():=> r'(t) > 0 V t e [ 0 ; - |. 2(a^ +b-^) + ab = (a + b)(ab+ !) 2(a' + b - ) + ab = (a + b ) ( a b + l ) 3 a > 0; b > 0 a > 0; b > 0 / nen i d ) la hiim dong bien trcn t e |{); - ] . . !. ^ , i ,, , 01 a = 2;b = l " ' " ' oc a = l;b = 2 Tir do suy ra 1(1) > 1(0) = 2 V t e (0; ^ ]. iH 23 V a y minP = — ^ khi va ch'i khi a = 2, b = 1 hoac a = 1, b = 2. Da Nhif the ta eo M > 2 V I e |0; ^ \ hi Binh ludn: nT M =2khi va ehi khi ab = be = ca; ab + be + ea = 0; a + b + c = 1 tuTe la khi va uO V i c e di/a P ve dang ( I ) la Ic liT n h i c n . Cai kho la t i m m i e n xac dinh cua bien ehi khi (a; b; e) la mot trong eac bo so ( 1 ; 0; 0), (0; 1; 0) va (0; 0; 1). D o do ie gia tri nho nha't ei'ia M la 2. iL b a Binh ludn: V i c e x c t dai lu'dng phu thu()e vao bien ab + be + ea la mot y nghia Ta Bang each kc'l hdp khco leo giffa dieii kien va bat dang thiife Cosi la suy ra hoan toan tiT n h i c n . D i c u do dan den vice x c t eac he thuTc (1) va (2). D e n s/ d i c u k i c n (6). Con l a i d l nhicn la silr dung phu'dng phap ehicu bien t h i c n ham 1 day vice xet ham so': r ( t ) = t " + 3t + 2 V l - 2 t vdti 0 < t < - up so" ham so dc giai bai loan. ro B a i 3: Cho eac so' thi/e khong am a, b, c thoa man a + b + e = 1. va suf dung phiTdng phap chieu bien thicn ham so de giai b a i toan la mot /g T i m gia I n nho nha't cua bicu ihtfe: / om vice lam tiT n h i c n ! Cach giai vifa trinh bay ro rang la each giai l o i U\ nha't. M = 3(a'b^ + b ' e ' + e'a^) + 3(ab + be + ca) + 2 Va^Tb^TiJ . B a i 4: (De thi tuyen sink Dai hoc, Cao clan,^ khoi D - 2010) .c (De thi tuyen sinh Dili Iwc, Cat) dcinii; khoi B ~ 20J0) T i m gia t r i nho nha't cua ham .so y = + 4 x + 21 - V - x ^ + 3 x + 10 trcn ok HUcing ddn giai mien xac dinh eija no. bo Ro rang ta CO bat ddng thiJe hien nhicn sau .t HuAng ddn giai ' " ce 3 ( a ' b ' + b'c^ + c'a') > (ab + be + ca)". " (1) H a m so' xac dinh k h i thoa mfin he sau: '» fa Ta l a i eo a^ + b ' + e' = (a + b + e ) ' - 2(ab + be + ea) = 1 - 2(ab + be + ca). (2) w. -x^+4x +21>0 f-3 (ab + be + ea)^ + 3(ab + be + ca) + 2 ^ 1 - 2 ( a b + be + e a ) . (3) Da'u bang Irong (3) xay ra k h i va chi khi c6 dau bang trong (1), ttJc la k h i va V i ( - x ' + 4x + 2 1 ) - (-X- + 3X+ l()) = x + 11 > 0 , (suy t u r ( l ) chi khi ab = be = ca. . , vay y > 0 V - 2 < X < 5, Dat t = ab + be + ea. Ta eo 0 < t < -(a + b + c)^ _ 1 Ta CO y- = (-x^ + 4x + 2 1 ) + (-X' + 3x + 10) - 2 7(-x^ + 4 x + 21)(x^ + 3x + 10) 3 ~ 3 " = (X + 3)(7 - X) + (X + 2)(5 - X) - 2 V(x + 3 ) ( 7 - x ) ( x + 2 ) ( 5 - x ' ) TCr (3) suy ra x c t ham so sau: f ( t ) = t^ + 3t + 2 N / 1 - 2 I vc'Ji 0 < t < ^ = (x + 3 ) ( 5 - x ) + (x + 2 ) ( 7 - x) + 2 - V(x + 3 ) ( 7 - x ) ( x + 2 ) ( 5 - x ) T a c o : f ' ( l ) = 2t + 3 + r(t) = 2 -
Chuy§n dg BDHSG JoAn gia tri Idn nha'l g\& trj nh6 nhS't - Phan Huy Khii Cty TNHH MTV DWH Khang Viet Tur (2) suy ra > 2. D o y > 0, nen CO y> V2 , V X e [-2; 5] (3) V i the ta C O ( - x ' + 3x + 10)(2x - 4 ) ' - ( - x ' + 4x + 21)(2x - 3 ) ' = -51x^ + 104x - 29 va c6 bang xet da'u sau: Da'u bang trong (3) xay ra o (a + 3)(5 - x) = (x + 2)(7 - x) x = j . 1 87 X Tuf do suy ra miny = ^/2 cs> x = -51x^ + 1 0 4 x - 29 i + 0 - 0 + 1 Binh luan: TCf do ta C O bang xet da'u sau: 1. V i e c ph^t hien ra y > 0 V x e [-2; 5] la le tif nhien va dcfn gian. Tir do ta 3 87 / X 2 5 01 stjr dung tinh chat cd ban sau nay: 2 1 3 51 oc N e u f ( x ) > 0 V x e D , thi m i n f ( x ) = /minf^(x). T 0 + + + 1 1l- iH 0 + + + 1.^^^-^^^ y' V i e c phat hien ra (2) la dieu binh thiTcJng v i chi dung eac phep bie'n d o i Da — — 1 1' ddn gian ve can thiJc da hoc kT d cap 2. y hi f \. PhU'dng phap bat dang thiJc ap dung d day tuy ddn gian nhU"ng c6 hieu qua IOV2 7A/2 nT Nhifvaytaco min y = y .3. 1 'tU Idn. Theo chiing toi do la each giai hi^u hieu nhat doi v d i b a i loan nay. 3 3 uO -2 0; 3 - 2x < 0. , ww ' . khi - < X < 1 2 ^ 2 u 1 2 ^ 4 (2) 3 " -X + x + y >2 ^ x ( x + y ) hay -x + x + y > ^ 2 x ( x + y ) T r d n c d s6 66 ta c6 ngay T > 0 k h i - < x < 2, va 1 2 Da'u b^ng trong (2) xay ra o - = o 2 x = x + y o x = y. ( 4 - 2 x ) V - x ^ + 3x + 1 0 - ( 3 - 2 x ) V - x ^ + 4x + 21 k h i - 2 < x < - X x+y T = 2 L a i theo ba't dang thtfc Cosi, ta co: yjlxix + y) < = ^— • (2x - 3)V-x^ +4X + 21 - (2x - 4)V-x^+3x + 10 khi 2 < x < 5. Difa vao nhan x 6 t sau neu A > B > 0 t h i A > B o > Tiif do thay vao (2) va c6: - + > — ^ (3) X x+y 3x + y
Cty TNHH ft/ITV DWH Khang Vigt :huy6n 6i BDHSG ToAn g\i lr| Idn nhS't va gi^ tri nh6 nhS't - Phan Huy Kh4i Da'u bang Irong (3) xay ra 2x = x + y x = y. M^l f,^ 251 25 )• = max < 12; — ^ = — max 1(1) = max j 1(0); r Tir(l)(2)(3)suyra A > — ( 4 ) 3x + y 191 Da'u bang Irong (4) xay ra o dong th(li c6 da'u bang trong {1), (2), (3) x = y. NhiT the minS = — dal difdc Thco gia ihic't ihi 0 < 3x + y < 1, nC-n lir (4) co A > X. (5) 16 2 + S x - y i x + y =1 X = Dau bang trong (5) xay ra o o X= y = — 4 3x + y =l V 4 1 « / 16 xy = — 2-V^ 01 16 X = NhiTvay m i n A = Xx = y = - . • . ' 4 4 .....W^M oc 4 X +y =1 iH lihih liiqn: PhiTdng phap ba't dilng thu'c to ro siJc manh ciia no trong lc(i giai 25 1 — i 1 (x;y) = Da noi trcn. 1 4 ^2"2J hi Jai 6: (De thi tuyen s'mh Dai hoc Cao dcfni^ kiwi D) liiiih liigii: Ro rang phufitng phap chicu bicn thicn ham so la thich hdp nhal dc nT Cho a i c so thifc x. y khong am va thoa man x + y = 1. T i m gia tri Idn nha't va uO giiii bai loan nay. nho nha't ciia b i c u thuTc S = (4x- + 3y)(4y- + 3x) + 25xy. ie Bai 7: (Dc thi tuyen sinh Dai IJOC Cao dan}' khoi B) HUiiitg dan giai iL Cho Ccic so ihiTc x, y lhay ddi thoa man dicu k i c n (x + y ) ' + 4xy > 2. Ta CO S = 16x'y' + 12(x'' + y ' ) + 34xy = Ifix'y- + 12(x + y)(x" - xy + y") + 34xy Ta T i m gia Irj nho nha't ciia bicu ihiJc A = 3(x"' + y ' + x ' y ' ) - 2(x- + y ' ) + 1. = 1 6 x ' y ' + I2(x + y ) | ( x + y)" - 3 x y | + 34xy. s/ HUiing ddii giai Do X + y = 1, ncn la CO S = 16x'y' - 2xy + 12. (1) '- up Dat xy = I. ... - V i c t lai A d i / d i dang A = ^ ( x ' * + y ^ ) + | ( x ' ^ + y-*) + 3 x - y - - 2 ( x - + y " ) + l ro /g Dox>();y>();x + y = 1 ncn{) 4xy, ncn lir gia ihict (x + y)' + 4xy > 2, suy ra f>^\ 25 (x + y ) ' + (x + y ) ' > 2 - » ( x + y ) ' + (x + y ) - - 2 > ( ) v I'd) 2 (X + y - 1 )|(x + y ) ' + 2(x + y ) + 2] > 0 (4) 16 Do (X + y ) - + 2(x + y) + 2 = l(x + y) + 11- + I > 0, ncn lir (4) c6: 1_^ 191 V a y min l(t) 1' x + y - 1 >()hay x + y > 1 > ; ()
Cty TNHH MTV DWH Khang Vigt Chuyen dg BDHSG Join gia trj I6n nha't \ii g\A trj nh6 nha't - Phan Huy KhSi nha't va nho nha't cua h a m so', l / u the cua phiTdng phap sijr dung bat d i n g thufc Tif d6 suy ra \6t h a m so f(t) = ^ t ^ - 2t + 1 v d i t > ^ . ro net qua thi du nay. ' - ^ ' ; (De thi tuyen sinh Caoddn}> khoi A) • ii-; ,v > * > ; T a c 6 f ' ( t ) = - t - 2 > 0 V t > - . N h i r v a y m i n f ( t ) = f' v2. 16 Cho x, y la hai so thi/c thoa m a n x^ + y^ = 2. T i m gia trj Idn nha't va nho nha't cua bieu thiJc P = 2(x^ + y^) - 3xy. TH do suy ra (diTa vao (3)): A > (6) Hudng dan giai 16 / Ta co: P = 2(x + y ) ( x ' + y ' - x y ) - 3xy = 2(x + y)(2 - x y ) - 3xy. (1) 01 1 D e thay dau bang Irong (6) xay ra o oc x=y o x =y=— Ta c6: x ' + y^ = 2 (x + y)^ - 2xy = 2 xy = ^ ^ ^ ^ ^ ~ ^ • (2) iH 1 Thay (2) vao (1) r o i dat t = x + y, ta c6: Da t = - .2 9 1 2 t^-2^ _ 3 ! l _ l =-e-^t^ + 6t + 3. , , .«nU;3w,;;:M''..^J.- hi N h i / t h e ta CO m i n A = — o x = y = —. P = 2t 2 - nT 16 ^ 2 • A 7. uO Bi?ih ludn: V i e c diTa ve xet bieu thtfc d o i v d i x^ + y^ la mot le tiT nhien. Bay g i d ta t i m m i e n xac dinh cua t: ie ,2 Bang phu'dng phap bat dang thtfc ta di den danh gia (3). 2 2.., (x + y ) 2 = x' + y > — — >(x + y ) ^ < 4 c : > - 2 < t < 2 . iL Cong viec con l a i di nhien can den viec van dung phiTdng phap chieu b i e n Ta thien ham so'. B a i toan quy ve: T i m gia tri Idn nha't va nho nha't cua ham so: s/ Ta thay viec ke't hdp nhuan nhuyen giiJa hai phifdng phap bat d i n g thiJc va f(t) = _ t 3 _ | t ' + 6t + 3 v d i - 2 < t < 2 up chieu bien thien ham so da dan den siT thanh cong trong qua trinh giai bai ro toan nay. Ta CO f ' ( t ) = - 3 t ^ - 3t + 6 va CO bang bien thien sau: ^,, , ^ 1 /g Bai 8: (De thi tuyen sinh Dai hoc Cao ddn^ khoi B) Cho X , y la cac so' thifc thoa man dieu k i e n x^ + y^ = 1. t -2 1 2 om ' 1 ^ .c f'(l) 0 ~ _ T i m gia t r i Idn nha't va nho nha't cua bieu thtfc P = ^^^^ ok l + 2xy + 2y^ i bo T \ Hitdng dan giai f(t) ce X e m Idi g i a i (ba each) trong bai toan 3, muc § 1 , chiTdng 1 cua sach nay. fa Binh luqn: B a i toan c6 den 3 Idi giai khac nhau (hai each suT dung phiTdng phap VaymaxP= max f ( t ) - f ( l ) = •5:^; w. m i e n gia t r i h a m so trong do c6 ke't hdp ca phiTdng phap li/dng giac hoa, mot -2
Cty TNHH MTV DWH Khang Vigt Chuy6n 6i BDHSG loin g\ tri I6n nhS't va gia tri nh6 nha't - Phan Huy Kh^i v i i C O b a n g b i c n i h i c n sau: 1. V i c e quy v c b i c u ihuTc d o i v d i x + y ( m a la sc d a i b a n g I ) r o i stir d u n g t) 1 I p h i f d n g p h a p e h i c u b i c n i h i c n h a m s o ' d c g i a i la v i c e l a m u / n h i c n ( g i o n g 0 r'(i) 1 - nhi/ irong nhicu bai loan x c l tri/c'tc day) 2. Ta ihur suy n g h l d i c u k i c n x~ + y " = 2 g(;i y la eo n c n siir d u n g p h i f t t n g p h a p i'(i) " l i M n g g i a c h o a " hay k h o n g ? > V a y m i n 1(0 = 1(1) = ^ V l X" + y " = 2, n c n d a i \= yfl sincji; y = sfl cos(p / • l>0 01 L i i e n a y P CO d a n g sau: , . , , , , , X~ 1 3 y^ I 3 /.^ 1 3 TCr do suy ra V oc X > 0, y > 0, /. > 0, la c6: — x 2 2 y 2 2 /. 2 P = 4 v'2 ( s i n \ + c o s \ p ) - 6sin(peos(p iH (sincp + cos(p)( 1 - sinci^coscp) - 6sintpcos(p Mi (4) TiT (3) suy ra S > - V Da X > 0. y > 0, / > 0. .- , . , - • (sincp f cos(p)" - 1 hi A p d u n g c i ) n g Ihifc smtpcoscp = — Dau bang Irong (4) xay ra x = y = z = 1. y; , ; ;, , , ; ' nT uO va d i l l I = sincp + eosip - V2 cosj c/(?/;,^' klioi B) .c T h c i t vay (3) x ' - 3x + 2 > 0 (x - l)(x' + x - 2) > 0 ok Cho X, v, / la ba so dirt
Cty TNHH MTV DWH Khang Vi$t Chuy6n dg BDHSG Jo&n gii tr| I6n nhflt g\i trj nh6 nha't - Phan Huy Khi\ Jy^ 1 ne'u y > 2 1 1 1 1 1 . . 2y .+ 1 neu y > 2 r + O - + - = — + — . ' (1) i + y' X y xy • i ^. (3) f'(y)- D a t X - - ; Y = - k h i do (1) c6 dang: X + Y = - XY + YI (2) 2y r-1 ne u y < 2 • ne'u y < 2 X y + y^ L u c nay A = X ' + Y ' = ( X + Y)(X^ - X Y + Y^). Ttf (3) suy ra f ' ( y ) > 0 V y > 2. V d i dieu k i e n (2) t h i A = ( X + Y ) l (3) K h i do bai loan da cho t r d thanh: Vl2y> ^/^7ne'u:^2+ N/3VX, y G R v a A = 2+73c>- up Cho X, y e R. T i m gia tri nho nhat cua bieu thuTc: .' ? I • ro /g A= V ( x - l ^^+y^ )' +7(x + l ) ^ + y ^ + y - 2 Tilf do ta CO mina = 2 + N/3X = 0, y = ^ . ; om • HUbng dan gidi .c Binh luqn: R i n g phep tinh vectd ta c6 danh gia (1). Sau do suf dung phiTdng L a y i i = (x - 1 ; y ) , V = ( - x - 1 ; y) => u + v = (-2; 2 y ) . ' ok phdp chieu b i e n thien h a m so de g i a i tiep bai toan dat ra. Theo phep tinh ve vectd ta c6: u + V hay bo B a i 1 4 : (De thi tuyen sinh Dai hoc va Cao ddnfi khoi B) v ce ^ ( X - l ) 2 + y 2 +J(x + l)2+y2 > 2 ^ 1 ^ ^ (1) T i m gia t r i I d n nha't va nho nhat cua h a m so f(n) = tren doan [ 1 ; e^]. fa Da'u bang trong (1) xay ra u, v la hai vectd cung phi/dng, ciing chieu X w. o X - 1 = - X - 1 X = 0. • ' v Hi/dngdan giai • ww T i r ( l ) s u y ra A > 2^1+ y^ +|y-2|. • •(2) x21nx.i-ln'x 21nx-ln'x lnx(2-lnx) Tac6 f'(x) = Dau b^ng trong (2) xay ra X = 0. x^ X' X e t h a m s o f(y) = 2 7 l + y^ + y - 2 vdiyeR. . ,.V?*V - ' D o x^ > 0 V X e [ 1 ; 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) 2^/r+y^ + y - 2 neuy>2 Ta CO f(y) = 2sjl + y^ + 2 - y neuy
Cty TNHH MTV DVVH Khang Vi?t X 1 x+ 1 D o x > - l = ^ x + 1 >()=>r(x) = > ( ) V x e 1 - 1 ; 2|. Inx i + + 1 2 - Inx _ i + 0 — L a i CO ! ' ( - 1 ) = 0. Tu" do suy ra min !(x) - 0 . y' + 0 - 1 4 , • (X + 1)" X- + l + 2x , 2x -' y ... Laiihay r ( x ) =^ = = \ —r- x- + 1 X- + 1 x^+1 1 0-^ 9 r Dol 2x < 1 f ' ( x ) < 2 =5. l"{x) < N/2 , V x 6 1 - 1 ; 2|. / x^ + 1 01 M a i khae 1 ( 1 ) = %/2 , iCr do suy ra max H\) = \f2 . oc Vcjy max R x ) - ("(c") = 4"; niin l"(x) = m i n i 1(0); r(c^)} = m i n j o ; ^1 = 0. -I
Chuyfin dg BDHSG loan g\i trj Idn nhat va gia trj nh6 nhift - Phan Huy KhSl Cty TMHH MTV D W H Khang Vi$t HUdng ddn gidi Tir do suy ra P > 8, vay minP = 8. Xem Idi giai trong bai loan 1, muc §1, chiTdng 1 cuo'n sach n^y. Cach giai nay sai d ch6 la mdi difa vao phan 1 cua dinh nghTa gid tri nho Binh luan: Day la mOt trong cac vi du dep nha't ve tinh da dang cua cdc each nhat. Ta xem phan 2 cua dinh nghTa c6 thoa man hay khong? De y r i n g da'u giai khac nhau dung de giai mot bai toan tim gia trj Idn nha't va nho nha't cua bing xay ra trong bat ding thuTc P > 8 k h i x = y = l . Tuy nhien khi d6 x^ + y^ ham so (bai toan c6 den 4 each giai khac nhau suT dung hau he't cac phiTdng = 2 (khong thoa man dieu ki$n x^ + y^ = 1). Vay khong the xay ra da'u bing phap cd ban de tim gia trj Idn nhat va nho nha't cua ham so: phiTdng phap bat trong baft d i n g thtfc P > 8, tuTc la phan 2 cua dinh nghTa ve gia tri nho nha't dang thuTc, phi/dng phap chieu bien thien ham so, phiTdng phap liTdng giac khong diTdc thoa man. . / 4 Hi (Hi SI 01 hoa, phtfdng phap mien gia tri ham so, phiTdng phap do thi va hinh hoc). Vi thS' ket luan minS = 8 la sai. oc Cdch giai dung nhif sau: Viet lai S diTdi dang: iH §3. C O s d LI THUYETCUA BAI TOAN 1 x 1 y S=l+x+-+-+l+-+y+- Da y y X X TIM GIA TR! LdN NHAT VA NHO NHAT CUA HAM SO hi f 1' f 1 ' X y 1 nT A. Dinh nghia gid tri Idn nhat va nho nhat cua ham so. x + — + y+zr- + - + - +— — + — + 2. (1) I 2x> [ 2 y ; ly ^ ) 2 U yj uO Dinh nghTa 1 : Xet ham so f(x) vdi x e D. Ta noi rang M la gia tri Idn nha't ciia ie f(x) tren D, neu nhiT thoa man cac dieu kien sau: Tac6: x + — >>/2; y + — >J2; (2) iL 2x 2y 1. f ( x ) < M , V x 6 D, , . . Ta 2. Ton tai Xo e D, sao cho f(X()) = M. (3) s/ Khi do ta ki hi$u: M = max f ( x ) . ,'ff.,ftu up XfcD Matkhac - + - > - ^ ro Djnh nghTa 2: Xet ham so f(x) vdi x e D. Ta noi r i n g m la gia tri nho nha't cua X y Vxy /g f(x) tren D, neu nhuT thoa man cac dieu kien sau: 1 1 om 1. f ( x ) > m V x e p , Do X + y^ > 2xy, nen tir (4) ta c6 - + - > • = 2V2.(5) y x2+y2 .c X 2. Ton tai X ( ) 6 D, sao cho f ( X ( ) ) = m. •( - 1< 5" V ok Khi do ta ki hieu: m = min f ( x ) . bo xeD Tir(l)(2)(3)va (5)suyraS>3N/2 +4. (6) ce Nhir vay dinh nghTa gia tri Idn nhat va nho nha't deu c6 hai phan. Can liAi y Dau bing trong (6) xay ra dong Ihdi c6 dau bing trong (2), (3), (5) fa r i n g ca hai phan nay deu quan trong nhu-nhau, khong diTdc xem nhe phan 2. w. Xet thi du sau day: o x =y= ww Cho X > 0, y > 0 va x^ + y^ = 1. Tim gid tri nho nhat cua hiiu thtfc sau: Nhir vSy ton tai (x,,; y«) thoa man xf, +y?) = 1 va S = 3^12+4 khi x = x,,; P = (l +x) + (l +y) y = y,). Theo dinh nghTa ve gia tri nho nhat, ta c6 minS = 3yl2 + 4 . yj Qua thi du nay ta thay neu khong de y den di^u kiC*n 2 trong dinh nghTa gid f y + 1^ X y tri Idn nhat v^ nho nha't cua ham so c6 the se dan den sai lam. Xet phep giai sau day: P = 2 + f X + — 1^ + - + — + - I x; k y) Taco x + - > 2 ; y + - > 2 v a - + ^ > 2 . X y y X 31
Chuy6n dg BDHSG ToAn gia tr| Mn nhift vji glii tr| nh6 nha't - Phan Huy KhJi Cty TNHH MTV D W H Khang Vigt 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)|>. (2) Tinh chfi't 1: Gia siJ f(x) xdc dinh tren D va A, B ia tap cua D, trong do A e B. xeD xeD) xeD2 i Gia sur ton tai max f(x); max f(x); min f(x); min f(x). i : . > > ;• v ; Chtfngminh: , . - xeA xeB xeA xeB Ta chi can chu'ng minh (1) (con (2) diTdc chi^ng minh b^ng mot each hoan Khi do ta CO max f(x) < max f ( x ) , (1) xeA xeB toSn tifdng lir). V i Di c D, I = 1, 2 nen theo tinh chat 2, ta c6: ,^ min f ( x ) > m i n f ( x ) . (2) niaxf(x) g(x) Vx e D. < , ; «j - mot trong hai tap D j , D2. Tif do c6 the cho la (ma khong lam giam sur tong uO Gia SIJT cijng ton tai max f(x); max g(x), khi do ta c6 m a x f ( x ) > m a x g ( x ) . quat)x„GD,. ie xeD xeD xeD xeD Tif Xo e D | nen theo dinh nghia ve gia tri Idn nhat, ta c6: iL Chi?ng minh: Gia suf max g(x) = g ( X ( , ) , vdi x„ e D. ' f(x„) g(x) V X e D =^ f(xo) > g(Xo). V', v.. s/ Do max f(x) > f(x,,) > g(x„) = max g(x) => dpcm. Hien nhien max f(x) < max< max f(x); max f(x)}-. (6) up xeD xeD [xeDi xeD2 ro Chii y: Menh de dao noi chung khong dung, ttfc la neu max f(x) > max g(x) Tif (5), (6) suy ra f(x„) = max f(x) < max max f(x); max f(x) /g (7) om xeD xeD] xeD2 thi chi/a the kct luan diTdc f(x) > g(x) V x e D. 1 .c Xet hai h^m so f(x) = 4 - x va g(x) = x Bay gid tir(4), (7) di den: max f(x)=:max-^max f(x); max f(x) xeD xeDi xeD2 ok tren mien D = {x: 0 < x < 3 } . bo => Do la dpcm. R6 rang max f(x) = 4 , ce () g(x) V x e D.. Xet mot vi du minh hoc sau day: Tinh chfi't 3: Gia SIJT f(x) xdc dinh tren mien D va D = Dj u D2. Chox>0, y>Ova x + y < 6 t^. Gia thiet ton tai max f(x), min f(x) V I = 1, 2. Tim gia tri Idn nha't cua bieu thiJc P = x^y(4 - x - y). xeD: xeD; DatD= {(x;y):x>0;y>0; x + y
Chuyen dg BDHSG Join gJA irj I6n nhat va g\A tr| nh6 nha't - Phan Huy KhSi K h i do ro rang D = D , u D 2 . Cho x > 0; y > 0 va X + y < 6. Theo nguyen l i phan ra (tinh chat 3), ta c6 : T i m gia t r i nho nhat cua bieu thuTc P = \^y(4 - x - y). K i h i c u D , D | , D2 nhif trong thi du minh hoa cua tinh cha't 3. * max P = max max P; max P (1) (x;y)eD (x;y)eD| (x;y)6D2 Theo tinh chat 2 thi m i n P = - max ( - P ) . ' (1) (x.y)eD (x,y)€D V d i m o i (x; y ) e D , thi 4 - x - y < 0 v i do x > 0; y > 0 nen P < 0 V (x; y) € D, Ta CO - P = Q = xV(x + y - 4). K h i (x; y) e D2 =^ X + y - 4 < 0 => Q < 0. L a i c < 5 ( 2 ; 2 ) G D| va k h i d 6 P = 0, ncn max P = 0. (2) Mat khac (2; 2) G D2 va khi do Q = 0 nen ta c6 max Q = 0 . (2) {x;y)eD| / (x;y)eD2 01 V d i m o i (x; y) e D2 thi 4 - x - y > 0, nen theo ba't dang thufc Cosi, ta c6: K h i do (x; y) G D | => x + y - 4 > 0, nen theo ba't dang tMc Cosi, ta c 6 : oc ^4 -]2 ^+-^- + y + ( 4 - x - y ) iH ^- + ^ + y + (x - y - 4) \ P = 4||y(4-x-y) X + y - 2 < 4). ; , ie Ro r a n g ( 2 ; 1) e D2=> max P = 4 . (3) Mat khac do (4; 2) G D , va khi do Q = 64, nen ta c6 max Q = 64 .(3) iL (x;y)eD2 (x;y)eD| Ta Tir(l)(2)(3)suyra max P - m a x { 0 ; 4 } = 4 . Tur (2) (3) va theo tinh cha't 3 (nguyen l i phan ra), ta c6 : s/ (x;y)eD , , max Q = max{();64) = 6 4 . (4) up 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 (x;y)€D , , I xeD ro Bay gid tCr (1), (4) suy ra mm P = - 6 4 . O.I' min f ( x ) . K h i do ta c6: max f ( x ) = - m i n ( - f ( x ) ) ; min f ( x ) = - m a x ( - f ( x ) ) . /g (x;y)€D xeD . xeD xeD xeD xeD om 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. ChuTng m i n h : Gia suT M = max f ( x ) . , .c X6D D a t t ( x ) = l|(x) + t2(x) + ...+ l„(x). ' ok ;M V x e D 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 ) vdi moi i = i . n . K h i do theo dinh nghla gia t r i Idn nhaft, ta c6: • ^^^^ ^ bo X6D xeD xeD xeD . [f(x„) = M,vdix„eD. ce K h i do ta c6: m a x f ( x ) < m a x f , ( x ) + maxf2(x) +... + m a x f „ ( x ) , (1) f-f(x)>-M VxeD xeD xeD xeD xeD fa TCf he tren suy ra !-f(x„) = - M . min r(x) > m i n f, (x) + m i n f2(x) +... + m i n t'n ( x ) . (2) w. xeD xeD xeD xeD ( ww Theo dinh nghia cua gia t r i nho nhat, tCf (*) suy ra m i n ( - f ( x ) ) - -M . Da'u b^ng trong (1) xay ra k h i va chi khi ton tai x„ G D sao cho NhiT vay ta d i den max f ( x ) = - m i n ( - f ( x ) ) =>dpcm. , , ... maxi;(x) = fi(x„),Vi = l . n . xeD xeD xeD , i Da'u bang trong (2) xay ra k h i va chi khi ton tai x„ G D sao cho Phan sau chiJng m i n h hoan toan tiTctng tiT. ' minf,(x) = fi(xo),Vi = l , n , Nhan xet: T i n h cha't 2 cho phep ta chuycn bai toan t i m gia tri Idn nhat thanh bai X G (J todn t i m gia t r i nho nhat hoSc ngmJc l a i . D i e u nhy c6 ich trong nhieu tri/dng ChiJng minh: hdp cu the se xet sau nay. ' Ta chiJug m i n h (1) (vdi (2) phep chilng minh hoan toan tu'dng tiT). X 6 t t h i du m i n h hoa sau day: ' * La'y tOy y X e D . Theo dinh nghia cua gia t r i Idn nhat ta c6 : 35 1A