Giá trị lớn nhất, giá trị nhỏ nhất - Chuyên đề bồi dưỡng học sinh giỏit: Phần 2

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Nối tiếp nội dung phần 1 tài liệu Chuyên đề bồi dưỡng học sinh giỏi - Giá trị lớn nhất, giá trị nhỏ nhất, phần 2 cung cấp cho người đọc các kiến thức: Phương pháp lượng giác hóa tìm giá trị lớn nhất và nhỏ nhất của hàm số, phương pháp chiều biến thiên hàm số, phương pháp miền giá trị hàm số,... Mời các bạn cùng tham khảo.

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Nội dung Text: Giá trị lớn nhất, giá trị nhỏ nhất - Chuyên đề bồi dưỡng học sinh giỏit: Phần 2

Chuy6n dg BDHSG Toan g\i trj Idn nha't va gia trj nh6 nhat - Phan Huy KhSi ^ P>2 Vay minP = 2 X = y = z > 0. .• ,., 0,^^ PHUaN6PHliPllf9N6GttCHdA B a i 42. Cho x, y, /. > 0 va ihoa man xy/ - 1 TiMGlATRIltfNNHKtlNi 1. T\m gia trj nho nhat cua P = (x + y)(y + /.)(/. + x) - 2(x + y + /). fx + y ly + z Iz + x NHiNHlttCdAHAMStf 2. Tim giii tri nho nhat ciia hicii thiJc Q = . + + , V x + l \ y + l V z + l LiTdng gi^c h6a I I m o t trong nhi^ng phrfdng phdp hay suT dung de t i m gia t r i / ffUihtgddn gidi ^Kt j :^>f? '' 01 „ /V . . . y . Idn nhaft, b6 nhaft cua h a m so'. oc 1. A p dung dong nhat thufc: B^ng phifdng phap d d i bien lifdng giac (thi du x = sint, x = cost h o l e x = + y)(y + z)(z + x) = (x + y + z)(xy + yz + zx) - xyz. (*) tKt" iH (X tant,...) ta diTa bieu thiJc va dieu k i e n cua bai todn ve dang luTcJng gidc. Tir d6 Da Ta c6: P = (x + y+ z)(xy+ yz + z x ) - x y z - 2 ( x + y + z ) . ,.>,Vr* d) difa vao phep tinh Itfctng giac ta se de dang hdn trong trong viec g i a i b a i toan hi T h c o ba't dang thiirc Cosi, ta c6 x + y + z > 3 ^/xyz = 3 (do xyz = 1 ) (2) tim gia t r i Idn nhat, nho nhat da cho ban dau. nT Cic bai toan t i m gia t r i Idn nha't, nho nha't c6 the suT dung phtfdng phdp liTdng L a i CO xy + yz + zx > 3 ^ / x V z ^ = 3 (do x^yV^ 1) (3) uO gidc hoa thi/dng c6 cac dau hieu de nhan bie't sau day: ie TCr ( I ) (2) (3) suy ra P > 3(x + y + z) - 1 - 2(x + y + z) - HoSc la trong bieu thtfc cua d a i liTdng can t i m gid t r i Idn nhat, nh6 nha't c6 iL => P > (X + y + z) - 1 > 3 - 1 = 2. (4) chlfa cdc d a i liTdng dang x^ + y^; 1 + x^;... Ta De thay dau bang trong (4) xay ra x = y = z = 1. ,;. : - Ho$c la dieu k i e n trong bai toan ban dau c6 dang: x^ + y^ = a^ a > 0,... s/ 2. TrU'dc he't ta chii'ng minh rhng - HoSc la cdc bieu thtfc da cho ban dau g^n lien v d i m o t h? thtfc liTdng gidc up (x + y)(y + z)(z + x) > (X + l)(y + l)(z + 1) (3) quen biet nao do. ro That vay dufa vao (*) suy ra /g B j k i l : C h o x , y , z G [ 0 ; 1]. (5) (x + y + z)(xy + yz + zx) - xyz > xy + yz + zx + x + y + z + 1 \ om T i m gia t r i Idn nha't cua bieu thiJc P = ^ x y z + J ( l - x ) ( l - y ) ( l - z ) . " o (x + y + z)(xy + yz + zx) - 2 > xy + yz + zx + x + y + z + 2 (do xyz = 1) .c o(x + y + z)(xy + yz + z x ) > x y + yz + zx + x + y + z + 3. (6) Htidng ddn giai ok Do x, y , z G [ 0 ; 1 ] , nen dat x = sin^A, y = sin^B, z = sin^C, bo D o xyz = 1 = > X + y + z > 3 va x y + yz + zx > 3. (7) ce Ta CO (x + y + z)(\ + yz + zx) ^ vdi A, B, C e zx) fa x +y+z, , , , x y + yz + zx (x + y + z)(xy + yz + = ^ ( x y + yz + zx) + (x + y + z ) - ^ - 4 + r—^ ' w. K h i d 6 0 < s i n A < l ; 0 < s i n B < l , 0 < s i n C < l ; 0 < c o s A < 1; ww («) 0 < c o s B < l , 0 < c o s C < 1. ! ' ' r'I Tilf (7) (8) suy ra (x + y + z)(xy + yz + zx) > x y + yz + zx + x + y + z + 3. Ldc nay ta c6: P = Vsin^ A s i n ^ Bsin^ C + Vcos^ A c o s ^ B c o s ^ C ' V a y (6) dung, tuTc (5) dung. = sinAsinBsinC + cosAcosBcosC. ML'l; . ( (1) (x + y)(y-f-z)(z + x ) Ta c6: cosAcosBcosC 33 ( (x + l ) ( y + l ) ( z + l ) • rz=o cosC — 1 Tir (5) (9) suy ra Q > 3. (10) t)a'u b^ng trong (2) x a y ra o o rx = l (*) cosAcosB = 0 D a u b^ng trong (10) xay r a o x = y = z = l . LLy = l Vay minQ = 3 o x = y = z = l . TtTdng tuf SinAsinBsinC < sinAsinB. O)
Chuy6n gj BDHSG Toan glA trj Idn nhat va g i i tri nh6 nhS't - Phan Huy Khii Cty TNHH MTV DWH Khang Vi$t 'z = l => tan^a tan^p tan'y tan^ 8 > 81 => x''y''zV > 81 => P = xyzt > 3. (7) sinC = l Dau b^ng trong (3) xay ra x=0 (**) Dau bSng trong (7) xay ra dong thcJi c6 dau b^ng trong (4), (5), (6) sinAsinB = 0 y =o a = P = Y = 8 Tir (1), (2), (3) c6: P < cosAcosB + sinAsinB =>P< cos(A - B). « x = y = z = t= ^ . (4) (8) Dau b^ng trong (4) xay ra o dong th5i c6 dau b^ng trong (2), (3). Vay min P = 3 o x, y, z, t thoa man (8). < ' . , .; ; V i cos(A - B) < 1 va dau b^ng xay ra khi va chi khi A = B, nen ta c6: ^hdn xet: Hoan toan tiTcfng tif, ta co ke't quS sau: - . , , , / 01 P 0 , y > 0 , z>Ovlk —^—- + —J—+ — + — ^ = 1. oc Dau bkng trong (5) xay ra o A = B dong thfJi thoa man (*) va (**) 1 + x^ 1 + y^ l + z" 1 + t^ iH X = y dong thdi thoa man (*) (**) Khi do neu P = xyz, thi min P = 2 72 . Da z - 0 ; x = 0;y = 0 Bai 3: (De thi tuyen sinh Dai hoc. Cao dann khdi B) (6) hi x=y=z=l Chox^ + y^ = i . nT ' Tir do ta CO max P = 1 o x, y thoa man (6). uO Tim gia tri Idn nha't va nho nha't cua bi^u thtfc: P = ^ ( ^ + 6 x y ) ^ Bai 2: Cho x, y, z, t > 0 va thoa man dieu kien: 1 + 2xy + y^ ie 1 1 1 1 • '(pp!< iL Hitdng din giai T + T + 7 + 7 =1 • Ta Dap so: max P = 3; min P = - 6 s/ Tim gia trj nho nha't cua bieu IhiJc P = xyzt. Xem IcJi giai trong bai toan 3, § 1, chiTdng 1 cuo'n sach nay. up -'^ Hudngddngidi Bai 4: (De thi tuyen sinh Dai hoc, Cao ddn^ khoi B) . • o ro Dat x^ = tana; y^ = tanP; z^ = tan y; t' = tan 8 vdi a, P, y, 8 e Tim gia trj Idn nhat \h nho nha't cua ham so: f(x) = x + \/4-x^ tren mien /g xdc dinh cua no. om Tilfd6: 1 + x" = 1 + tan^a = - 4 — ; I + y" = — 1 - • Hudngddngidi - .c cos'p CQS Qt Ddp so: max f(x) = 2>j2 ; min f(x) = - 2 ok 1 Xem Idi giai each 3 trong bai toan 1, bai 1, chiTcfng 1 cuon sach n^y. bo cos^ y cos^ 8 B^i 5: Cho x la so thifc tily y (x e R). ce fa VI vay dieu kien: + -r + j + j =1 Tim gid tri idn nhat va nho nhat cua ham so: f(x) = ^ + 4x + 3x w. • 1 + x^ 1 + y^ l + z^ 1 + (i+x^r ww o cos^a + cos^p + c o s \ cos^ 8 = I (1) HUdng ddn giai Tir (1) suy ra: sin^a = cos^p + cos^y + cos^6 (2) E>^tx= tancp vdi x e Khi 66 ta c6: ' Ap dung bat dang thiircCosi, ta c6: sin^a >3^/cos^8cos^cos^y . (3) "2' 2 Lap luan tiWng tif, ta c6: (; sin^p > 3^cos^acos^8cos^^ y , (4) 3 + 4x^+3x^ 3 + 4tan^(p + 3tan'*(p , , ^ 4 \ — "2— = 5—21=:(3 + 4tan''(p + 3tan^(pjcos 9 sin^y > 3^os^ a cos^ Pcos^ 8 , (5) ( l + x^) ( l + tan2(p) sin^ 8 > 3^/cos^ a cos^ pcos^ y . (6) i - - s i n ^ 2 ( p + sin^2(p = 3cos'*(p + 4sin^(pcos^(p + 3sin'*(p = 3 \ ' 2 ' • Nhan tiTng vd' (3), (4), (5), (6) va c6: = 3-—sin^2cp. (1) sin^asin^PsinVin^ 5 > 8 Icos^a cos^p cos^os^ 8
CtyT.lli M i V DVVH Khang Vi$t ChuySn BDHSG Toan gii trj Idn nha't g\& tr| nh6 nha't - Phan Huy Khii (8x + 12x'')(l + x ^ ) ^ - 4 x ( l + x^)(3 + 4x2+3x'*) Tir do: f'(x) = 7C n X6t ham so F((p) = 3 - ^sin^ 2(p, vdi cp e (l + x '2''2) ^(8x + 12x-'')(l + x ^ ) - 4 x ( 3 + 4x^+3x^) 4x-^-4x _ 4x(x^ - 1 ) Ta tha'y ngay: min F(cp) = 3 - ^ = | « si 2(p = 1; (l + x^)' ~ ( l + x2)'~ (i + x ^ r maxF((p) = 3 - 0 = 0 o s i n ^ 2 ( p - 0 . Vay C O bang bien thien sau: " '^'^^ •' -00 -1 0 1 +00 / xeR 01 2x VXGR =>(X^+1)^ >4X^ VxeR. Vay m = 3 la mot gia tri ciia f(x). ce J • Neu m 9^ 3, khi do (6) c6 nghicm khi va chi khi phiTdng tnnh sau (an t): fa Tir do theo (2) suy ra: — | V x e R . • (4) Ta c6 nhan xet vi - = - ! I l ^ = l , nen neu (7) c6 nghicm thi hai nghiem Dau bang trong (4) xay ra o x^ = 1 o x = ± 1. a m-3 phai cung dau Tir do va theo (5) suy ra: min f(x) = | x = ±1. Do do (7) C O nghiem khong am khi va chi khi > ; ;l xeR 2 A'>0 2m-5>0 Ta thu lai ket qua tren. 2(m-2) » i m-2 o —
Chuy6n BDHSG Toan g\& trj Idn nhS't giA trj nh6 nhS't - Phan Huy KhSi CtyTNHH MTV DVVH Khang Vi§t Tu'(5)suyra: max P = 2 V 2 - 2 , (6) Ket hdp lai (6) CO nghiem - < m < 3 . (x;y)eD2 min P= -2V2-2. (7) Tir do suy ra: max f(x) = 3; min f(x) = ^ . , ,, (x;y)eD2 xeR X£R 2 Ttf (3), (6), (7) suy ra: max P = max [ijl - 2; o | = 2^2 - 2 , Mot Ian niJa cac ban thay diTOc tinh da dang cua cdc phUWng phap giai bai (x:y)eD toan gia tri Wn nhat, nho nhat cua ham so. , j min P=:min|-2V2-2;0|=-2N/2-2. (x;y)£D Bai 6: Cho x va y la hai so thifc khong dong nhat bKng 0. / Sh$nxet: 01 x^ — (x — 4y)^ j DT nhien ta c6 cac ciich giai khac nhau nhifsau: oc Tim gid tri Idn nha't va nho nha't cua bieu thiJc: P = : 5—. ' x^ +4y'^ .: iH N HUdng ddn giai \ Dat — 2y = t, t e R. luc do xet ham so Rt) = ' l + t^ = 4 t ^' + ~ 1* Da Dat D = { ( x ; y ) : x ^ + y ^ >0J 0. Tim gia tri Idn nha't va nho nha't ciia bieu thuTc: 1 + lan^ g p _ (x-y)(l-xy) . = 2sin2g - 2cos2g - 2 = 2N/2 sin -2. (5) (l + x ) ' ( l + y 2 ) ' 4/ v "I .!.•/• y 197 196
Chuv6n dg BDHSG Join g\i tr| Idn nha't va gia trj nhi nhSt - Phan Huy KhSi Cty TNHH MTV DVVH Khang Vi?t Hudng ddn gidi x-'+y^ xy sin^a + cos^ a sln^ a cos^ a Ddn ^o; max P = —; min P = I ' 4 4 1 1 4 Xem 151 glal (bkng phifdng phap liTdng gidc hoa) trong bal toan 1, § 1, chiftJng 1 . (1) l-^sln22a -sln22a 4 - 3 s l n 2 2 a ' sln^2a I cuon sach nay. 4 4 Bai 8: (De thi tuyi'n sink Dai hoc, Cao ddn)^ khdi D) Datt = sin^2a.DoO< a < - = > 0 < 2 a < 7 i = > 0 < t < l . , Tim gla tri \6n nha't va nho nhat cua ham so: 2 ^ : • / 01 f(x) =-^iL trendoan[-l;2]. H(,..,, , Tir (1) dan den xet ham so f(t) = + - - ^ ^ ^ ~ ^ ] vdl 0 < t < 1 4-3t t 4t-3t^ oc Vx^ +1 iH ' i HUdngddngidi - ' ' - ' 8(-3t^ + 1 2 t - 8 ) •f'(t) = Da (4t-3t)^ Dap so: max f(\) = yl2; min f(x) = 0 ; hi -l X = sina, y = cosa, vdl a e [0; 2n]. sach nay (dung phiTdng phap li/dng giac hoa). Do u^ + v^ = 1 => u = cosp, y = sinp, vdl p 6 [0; 2Tt]. Bai l l : C h o x > 0 , y > O v a x + y = 1. ?! Vi the P = slna(cosp + slnp) + cosa(cosP - sinP) Tim gia tri nho nhat cua bleu thiJc: P = — j- + — • = sinacosp - sinPcosa + cosacosP + slnaslnp HUdngddngidi ,| = sin(a - P) + cos(a - P) = v^cos (1) Do X > 0, y > 0 va X + y = 1, nen dat X = sin^a, y = cos^a vdi 0 < a < | . Tir (1) suy ra: max P = N/2 va min P = - 72 . '' Luc nay: ' I 199
Chuyen BDHSG Toan g\& trj Ifln nhft vk gii trj nh6 nhit - Phan Huy KhSi Cty TNHH MTV DWH Khang Vi«t Nhan xet: Ta c6 the suf dung phiTdng phap dung bat d i n g thiJc Bunhiacopskj ^^f- ^'^"f each giai " p h i lUcfng giac h o a " sau day: de giai nhuTsau: v / T a c o : x ' + y ' = (x + y ) ( x ' * - x V + x ^ y ^ - x y ' ' + y ' ' ) Theo bat dang thiJc Bunhiacopski, ta c6: [x(u + V) + y(u - v ) f < (x^ + )[(u + v)^ + (u - v)^ ] , = ( X + y)[{x' +y^f-xY -xy(x2 + y^) h a y P ' < 2 ( x 2 + y ^ ) ( u 2 + v ^ ) - 2 (Do x^ + y ' = + = 1). = (X + y ) [ l - x y - x ^ y - ] , . . TClfd6tac6: -N/2 < P < V 2 . : " x ' + y^ = ( x + y ) ( x ' - x y + y2) = (x + y ) ( l _ x y ) . > >/2 A p dung cong thiJc: / C6 the tha'y P = - J l chang han k h i x = y = u = v = 01 2 -,2 ^ (x + y ) ^ - ( x ^ + y ^ ) (x + y ) ^ - l (x + y r - 1 oc (Ldc do ro rang: x^ + y^ = + = 1). xy =>x^y^ = iH P = - ^ c h i n g han k h i X = %/2 ; y = N/2 ; u = v = - Da T a c o : |P| = |l6(x5+ y 5 ) - 2 0 ( x - V y - ^ ) + 5(x + y)| va dat x + y = z, ta c6: hi Tijf do ta c6: max P= s f l ; m i n P = - y I l 2'! z^-1 (z^-l) nT 16z -20Z + 5z C(if hay tU tint xem khi nao P dat ^id trf Idn nhdt (nhd nhdt)? 4 ; 2 J uO B a i 13: Cho x > 0 , y > O v a x + y = l . T i m gia t r i nho nhat ciaa bieu thvJc: ie 5\ -47/ + 10z^-5 2z'-- (*) iL P = Ta Hitdng dan giai ^^^^^^^^^^^^ ' '1 V i x ' + y^ = 1 x ' + y2 + 2xy < 2(x2 +y2) = 2 =^|z|y^cos ' 5 a (3) 4 Hu^ng ddn giai Ttf (3) suy ra: max P = V2 ; m i n P = - A/2 . T a c 6 : x z + - + - = 1. (I) y y 200 201
IF Chuyen d6 BDHSG Toan g\i t r j lan nh^t g\A t r j nh6 n h f t - Phan Huy KhSi Cty TNHH MTV D W H Khang Vigt Do X , y, z la cac so diTdng, T. 10 1 n e n d a t x = t a n - ; - - t a n ^ ; z = t a n J , d day a . p . y e Tiif do suy ra: max P = — x = -j= ; y = V ^ ; z = - i = . 2 y 2 2 V 2 2V2 ffh4n x^*' Phi'f^ng p h a p " l i T O n g g i a c h o a " to ro h i e u qua tren b a i toan n a y ! Khid6(l)c6dang: tan^tan^ + tan|tan^ +tan|tan^ = i (*) pai 16: C h o x, y , z e [ 0 ; 1 ]. T i m gia tri Idn nha't c u a b i e u thiJc: P = ( l + x2)(l + y2)(l + z2) + ( l - x 2 ) ( l - y 2 ) ( l _ z 2 ) . a P Y 71 do - + - + 4- = - 2 2 2 2 HUdng ddn gidi / 1 01 Taco: 1 + x^ = 1 + t a n ^ ^ = D a t x = t a n Y ; y = tan^; z = t a n ^ . D o x, y, z e [ 0 ; I ] =>a, p, y e 0 ; ^ oc cos 2 iH ^ Ta c6: c o s a > 0; c o s P > 0; cosy > 0, v i the h i e n n h i e n suy r a : 1 Da P 2 (1 + c o s a ) ( l + c o s P ) ( l + cosy) > 1 + c o s a c o s P c o s y . ; 1 +y^=: 1 + cor^ = hi I 03*0 bkng trong (1) x a y r a o c o s a = c o s P = cosy = 0x = y = z = l . 2 nT iDi/a v a o c o n g thtfc luung g i a c , ta c6: ' uO 1+ = 1 + tan^ r = 1-x 2 A (l-x^)(l-y^)(l-z^) ie 2 cos^-^ / (1) o 1+ 1+ 1+ 1 + x^ l + y^ 1 + z^ (l + x2)(l + y2)(l + z2) iL VayP = 2 c o s 2 - - 2 s i n 2 ^ + 3 c o s 2 ^ - l + c o s a - ( l - c o s p ) + 3cos2^ Ta o 8 > (1 + x^ ) ( l + y 2 ) { l + z^) + (1 - x^ ) ( l - y^ ) ( l - 7 ? ) hay P < 8. 2 2 ^ s/ I Da'u bkng trong (2) xay ra o da'u b^ng trong (1) xay rax = y = z = l . = 2 c o s ^ c o s ^ . 3cos^ I = - 3 s i n ^ I + 2 s i n I c o s ^ + 3 up Vay max P = 8 o x = y = z = 1. • ro r. . y 1. Y \ 1 2a-P Binh luqn: Di nhien cac ban c6 the l ^ m nhuf sau: f /g 2 . y 2Y a - P sin-—sin-cos-—— o t - P = 3 - 3 sin - — s i n - c o s — — = 3 - 3 --cos D o x . y . z e [0; l ] = > l - x ^ > 0 ; l - y ^ > 0 ; l - z ^ > 0 . a 2 2 om 2 3 2 2 3 2 ) 9 2 V i the hien nhien ta c6: .c \ . y 1 a - P 10 1-x^ 1-Z 2"^ ok = 3 ^ i c o s ^ ^ - 3 sin-—cos——- 1+ 1+ 1+ 2 3 2J l + x^ 1 + z^ bo ( l + x=)(l + y ^ ) ( l + z ^ ) ' a - P , ce Tir (*) suy ra max P = 8 o X = y = z = 1. cos = 1 fa Da'u b i n g xay ra 2 Tuy nhien sur dung (*) khong phai la dieu mk ai cung tha'y difdc, trong k h i sur . y 1 w. sin—= - . dung (1) va cong thiirc lifdng gidc thi Idi giai tiT nhien h d n ! . 2 3 ww ^ai 17. Cho x, y , z la ba so difdng thoa man dieu kien xy + yz + zx = 1. 2Y I 1 9 , I 7 -Y = -1= > t a n Y 1 - =—7==>z = — 1 2 8 2 2>^ 2V2 T i m gia t r i Idn nha't cua bieu thuTc P = 9 1+x^ Vi+y^ ViT7 Tiif (*) v d i a = P suy ra , Hiidng ddn gidi tan^ I+2 tan|tanl -1 = 0 «tan^ | + ;^^'"f ~ ^^^"^^^"1^72 ' A B C BSt X = tan — , y = tan — , z = t a n — v d i A , B, C e 2 2 2 ^' ^ ^' A B B C C A V a y dau b^ng xay ra k h i va chl k h i x = y = V2; z = . Tfirxy + yz + zx = 1 => t a n — t a n — + tan—tan — + t a n — t a n — = 1 2 2 2 2 2 2 909
Chuyen dj BDHSG Toan g\i tr| I6n nhft gii tr| nh6 nhSt - Phan Huy Khii Cty TNHH MTV DWH Khang Vi^t Af B B C y 1 tan— tan — + tan — 1-tan—tan— ; ;i ^ Ti/cfng t y ta CO - = < - (2) 2 2 2I 2 2) i+r .y + z y + x; z z A 1 - tan —tan — + (3) B — + — C 7iT7 2 z+x z+y " , V 2 ° , B ' C — 12 2. Da'u b i n g trong (2) (3) tiT^ng uTng xay ra o z = x; x = y. < 2 2 tan — + tan— tan / A fB (t\ 01 tan — = cot —+ — I2 Congtiirngve'(l)(2)(3)vac6P< | . ' ' (4) 2) oc 2 DS'u b^ng trong (4) xay ra o dong thcfi c6 da'u b^ng trong (1) (2) (3) iH ^A + ^ + C ^ ^ y O =>A + B + C=180". 2 Da /; v v ^ "2 " 2 0, y > 0, z > 0, nen dat x = tanA, y = tanB, z = tanC,vdi A , B , C e 2J /g om A B C 3 s i n — + s i n — + s i n — < — va dau b^ng xay ra k h i chi k h i A = B = C = 6O". Tir gia thiet x + y + z = xyz, ta c6 tanA + tanB + tanC = tanAtanBtanC .c 2 2 2 2 ^ ' tanA( 1 - tanBtanC) = - ( t a n B + tanC) ok Tird6suyramaxP= I o A = B = C = 60" bo tanA = - t a n B + tanC ^ + C)=> A + B + C = I8O". ,j Vr. I-tanBtanC ce A B C 73 t a n — = t a n — = t a n — = — Vay CO the coi A , B , C la ba g6c cua mot tam giac A B C . fa 2 2 2 3 w. 73 Ta c6 = ^ ^ " ^ f = tan^ A.cos^ A = sin^ A ' o x s y =z = ww 1 + x^ 1 + tan^ A 2 2 NHn xet: Ta c6 the g i a i b ^ i todn tren b^ng each "phi lifdng giac h o a " nhiTsau: TiTdng tir = sin^ B; = sin^ C . V a y P = sin^A + sin'B + sin^C DiTa v ^ o xy + yz + zx = 1, ta c6 • 1 + y^ 1 + z^ X ' X X Ta biet r^ng trong m o t tam gidc thl sin^A + sin^B + sin^C < ^ . Vl + x^ ^xy + yz + zx + x^ 7(x + y)(x + z) V x + y Vx + y Dau b^ng trong (1) xay ra o A = B = C = 60" X X Tijf do theo ba't d i n g thuTc Cosi, ta c6
_Cty TNHH MTV DWH Khang Vi? ChuySn dg BDHSG Toan g\& trj Ifln nhS't va gia trj nh6 nhjt - Phan Huy Khai %\. (De thi tuyen sink Dai hoc. Cao ddn^ khoi D - 2010) GAwcfn^4. PHIfONGPHtPCHlfUBlfNTHItNHJlMSdf Tim gia trj nho nha't cua ham so: y = V-x^ + 4 x + 2 i -V-x^ + 3 x + l() tren TiM GlU TR| itN NHlt YA NHA NHiTr COA HAM Sdf mien xac dinh cua no. HiAhig ddn gidi Cung v d i phiTctng phap bat d i n g thiJc, day la m o t trong hai phiTdng phap (Lcfi g i a i v ^ n tat, I d i giai chi tiet xem bai 4, §2 chiTdng 1 cuon sach nay) thong dung nhat de t l m gia t r i Idn nhat va nho nha't cua ham so'. M i e n xac djnh cua ham so la: - 2 < x < 5 / 01 B k n g each x e t chieu bien thien h a m so (ma thong thu'cfng ngi/cti ta hay si} • y• - ~ > ^ - x ^ + 3 x ^ - ( 3 - 2 x ) V - x ' + 4x + 21 oc dung phep tinh dao ham), sau do so sanh gia tri ham so' tai c i c d i e m dSc biet 2 V - x ^ + 4 x + 21.V-x^ + 3x - 1 0 iH (thong thifdng 1^ cac d i e m cifc dai, ciTc tieu, c^c d i e m dat biet n h i / cAc dau Tif do cd bang bien thien sau: /.ui^ ,, , Da mut cua cac doan th^ng xac dinh nen m i e n xac dinh cua h a m so dang xet, i hi cac d i e m khong ton tai dao ham...). TiJf ph6p so sdnh ay suy ra cic gia tri Idn X I 5 3 nT y y ) nha't, nho nhat phai t i m . y y' y^ - 0 + y uO y y y y y ie y y t §1. SUr D M N G TRl/C TIEP C H I E U B I E N THIEN HAM SO y y y . iL D E TlM G I A TR! LdN NHAT. NH6 NHAT / 1 \ I Ta Vay m i n y = y 3; s/ Nhiyng bai toan nay thufdng c6 dang ddn gian hoac la trifc tiep khao sAt chieu up bien thien h ^ m so' can t i m gid tri Idn nha't, nh6 nha't cho trong dau b ^ i , hoac Bki 3. (Di thi tuyen sinh Dai hoc, Cao dang khoi B) ro la thiTc hien mot phifdng phap doi b i e n ddn giSn d\ia Mm so can khao sat In^x Ttm gid trj Idn nha't va nho nha't ciia ham so: f ( x ) = tren doan i;e-^ /g ve dang dcJn gian v i thuan I d i hdn cho viec t l m gia t r i I d n nha't, nho nhat om X bkng phiTdng phap chieu bien thien ham so. HUdng ddn gidi .c Bai 1. (De thi tuyen sink Dai hoc, Cao daitfi khoi D - 2011) Khao sat trifc tiep f (x) va suy ra max f(x) = - i - ; m i n f ( x ) = 0 . ok l
Cty TNHH MTV DWH Khang Vigt Chuyen aj BDHSG Toan g\A trj I6n nh&t va gia tri nh6 nhS't - Phan Huy KhSt HUfiHg ddn giai dday: F(t) = t ' + 4(1 - t)^ . Khao sat tri/c tiep r(x), roi suy ra: min^f(x) = -2; _max^f(x) = 2>/2 . Ta c6: F(t) = 3t^ - 12(1-t)^ =-9t^ + 2 4 t - 12. Vay CO bang bien thien sau: (Xem Idi giai chi tiet cl bai 1 (each 2), §1 chtfdng 1 cu6'n sach n^y). If /p X y , t 0 f • ^ Bai 6. Cho X, y e [1; 2]. Tim gia tri Idn nhat ciia bieu thufe: ? = - + F'(t) - 0 + 0 / HUdngdangi&i F(t) 01 oc D a t t = - . D o x , y e [1;2] = > ^ < - < 2 = > ^ < t < 2 Nhi/vay: min f(x) = min F(t) = F (-. 3 . 4 iH , „ • y 2 y ^ ;,, Da -1
Chuy6n dg BDHSG To&r\ trj Idn nhj't va g\& tr\6 nha't - Phan Huy KhSi Cty TNHH MTV DWH Khang Vigt Ttf do theo (1) c6: m i n P = 2f = 2.1 = ^ , 2J 3 3 F'(t) 0 1 1 F(t) m a x P = 2 m a x { f ( 0 ) ; f ( l ) j = 2 m a x \ - ' - - } = 1- 2 1 1 x = 0;y=:l Vay m i n P = — < = > x = — ; y = —, max P = 1 o x = l;y = 0 3 2 2 / Nhir vay: max P = max F(t) = m a x { F ( l ) ; F ( 3 ) } - max (4; 1 0 } = 10, 01 . , l y = l - x , v a y P = + =2 , f 1t\ Va y max P = max f ( I ) = f(0) = 1; min P = min f ( t ) = f ok 2-x x+1 _x^+x + 2 14J 0
Chuygn 6i BDHSG Toan g\i tr| Idn nhit va gia trj nh6 nha't - Phan Huy Kh^i Cty TNHH MTV DWH Khano Vi5t x + y =1 x = l ; y = () m i n f ( x ) = ^ m i n F ( t ) = 7 m i n { F ( 0 ) ; F ( l ) } = ^ n { 0 ; 0} = 0 . Tif do suy ra max P = 1 t = 0 o 0 0 V x € 0 ; ^ , nen ta c6: 256 Ta 2 Dau bang trong (4) xay ra * ^ ,( s/ max l'(x) = max l"^(x), (1) up 2 tos^ X 2 3 73 , . ()
Chuyen dg BDHSG Toan gia tri Idri nha't g\i trj nh6 nhaft - Phan Huy Kh^i CtyTNHH MTV DWH Khang Vi^t Taco: f^(x) = 2 + (sinx + cosx) + 2^1 + (sinx + cosx) + sinxcosx . HUi'fng dan giai LcJi giai v^n tat nhU'sau: (Xcm giai chi tic't trong bai 5, nhan xet 2 cua (sinx + COSX) -1 Dufa viio cong thijfc: sinxcosx = , ta dat t = sinx + cosx chiwng3) . --r -//.-^vv'^. 4x(x^-l) (-V2
Chuy6n dg BDHSG Toan gia trj Idn nha't va glA trj nh6 nhat - Phan Huy Khii Cty TNHH MTV DWH Khang Vijt minf(x) = min{F(-sinl); F(sinl)} =min|-2sin^ l - s i n l + 2; - 2sin^ 1 + sinl + 2| ddayF(t)= - | t 2 + t + | . = -2sin^ l - s i n l + 2. Ta c6: F'(t) = -I + 1, nen c6 bang bic'n thicn sau: 1 3 3^/2 §2: Sii DUNG C H I E U BIEN THIEN HAM SO C O KET HOP THEM C A C PHUdNG PHAP KHAC F'(t) F(t) DE TlM G I A TR! LON NHAT, NHO NHAT / 01 rCf (4), (5) suy ra: Vdi nhffng bai loan phtfc tap hdn, lifdc do suf dung phiTdng phap chieu bic'n oc thien ham so'de tim gia tri i(5n nha'l, nho nhat nhiTsau: max f(x)= max F(t) = F ( 3 ) - 3 , iH - Trydc hct bang cac bai toan phu (thi du nhif su" dung cac baft dang thi?c Da -3
Cty TMHH M I V DVVH Khang Vijt Chuyen ai B D H S G Toan gia tr| \6n nhSt va g\A tr| nh6 nhS't - Phan Huy KhSi 2. Bang phu'dng phap giai hoan toan ttfdng tif, cac ban hay giai biii loan sau: DSthafy: - 2 . D a / 7 . w . m i n P = 2. u y X / 4 , 3 01 r(u) m - 3 • ' oc Bai 3. Cho x > 0 , y > 0 , z > O v a x + y + z < ^ . 17. iH f(u) • 4 ^ \ ^ 1 3 1 I l l Da Tim gia tri nho nha't cua bieu thu'c: P = x + y + z + - + - + - . 6 hi X y z '• nT m. 13 17 HUdngd&ngidi -.-^'^.i'*' Vay ta co mien xac dinh cua ham so: F(t) = t" - 5t^ + t + 4 la — < t < — . n uO 6 4 Theo ba't d i n g thu'c Cosi cd ban, ta c6 ' 1 1 ie Taco: F'(t) = 4 t ^ - lOt + 1 = 4 t ( t ^ - 4 ) + 6t + 1 . (x + y + z) + — + — + - (1) y z X y z x+y+z iL X NhirtheF'(t)>0khit>2. 9 Ta Tir(l)suyra: P > x + y + z + (2) Do — < t < — , nen noi rieng ta c6 bang bien thien sau: s/ 6 4 ^' up Dau b^ng trong (2) xay ra o x = y = z. a Jil;' H.Sw iM,.gii ro Datt = x + y + z = > 0 < t < - . X e t h a m s 6 ' : f(t) = t + - v d i O < t < - . /g om 2 t , .2,^,^ ^ , Ta c6: f ( t ) = 1--^= . va c6 bang bigVi thien sau: '^^ ^ .c I ok t -3 0 bo 4249 ' t v\ J Vay maxP= max F(t) = F ce v4 16 ' 1 > f(t) if fa f(t) 1083 w. minP= min F(t) = F f—1 .4, 54 i3 - + - = — o 16 y X 4 x = 4;y = l Tir(2), (3) suyra:P >y. (4) . 1083 X y 13 x = 2;y =3 ' v ^ r ''v . 'mmP = - + - = — x = y = z xitt'-'ts • . 54 y X 6 x = 3;y = 2* • v ' t / ^.^t. '^v. 1 Da'u b^ng trong (4) x5y ra 3 o x =y=z= - ,?
Chuyen 66 BDHSG Toan gia lr| Idn nhat va giei tri nh6 nha't - Phan Huy Kh^i _^ Cty TNHH MTV DVVN Khang Vi^t Nhan xet: Suf dung baft dang thuTc phu, ta c6 danh gia (2) HUefng ddn gidi T r e n ccf set (2) siir dung phifdng phap chieu bien thien h ^ m so' dc giai tie'p hl\\ X e m lc*i giai chi tiet trong bai 10, §2, chiTdng 1 cuo'n sach n^y. toan. D a y la su" k e t hdp giffa phu"dng phap bat dang thiJc va chieu bien thie,i pal dang khoi B) ham so de t i m gia t n nho nhat ciia P. Cho X , y , z lii ba so' thiTc difdng. T i m gia trj nho nha't cua bieu thiJc: B a i 4. (De thi tuyen sink Dai hoc, Cao ddnn khoi A-2011) fx 1 ^ 1 y • fz 1 ^ P = Xx 1 +y —+ — + •/. 1 Cho X > y , X > z va X , y , z e [ 1 ; 4 ] . T i m gia t r i nho nha't cua bieu thuTc: 12 yz; I 2 ZX y l2 xyj X y z ;s''"'^ P = X Hi/dng ddn gidi / 2x + 3y y+z z +x 01 X e m l(li giiii chi tiet trong bai 11, §2, chiTdng 1 cuon sach nay. oc ' • •• • . HUdng ddn gidi ' " ' ^'^''^ Bai 10, (De thi tuyen sinh Dai hoc, Cao dang khoi B) iH X e m Idi giai chi tie't trong b a i 4, §1, chi/dng 1 cuon sach nay. Chox, y e R. . „ , r;, f /V; Da Binh luan: D a y la sif k e t hdp kheo leo giffa hai phiTdng phap bat dang thiJc T i m gia t r i nho nha't cua bieu thtfc A = yj(,x-lf +y^ + + + y^ + |y - 2 hi va chi^u bien thien ham so de t i m gia t r i nho nha't cua P. Hii/ing ddn gidi '•*•.< f.\ nT B a i 5. (De thi tuyen sinh Dai hoc, Cao ddnfi khoi B-2011) X e m 15i giai chi tie't trong b a i 13, §2, chu^dng 1 cuo'n sach nay. j,( uO Cho a, b la hai so thifc diTdng va thoa man dieu k i e n : Binh luan: Trong cac de thi tuyen sinh noi tren (ttf bai 4 den bai 10) de giai cac ie 2 (a^ + b^ ) + ab = (a + b)(ab + 2 ) . bai toan t i m gia tri Idn nha't, nho nha't cua cac bieu thuTc deu diTa v^o suf k c t iL Ta^ \ ^2 hdp kheo leo giffa phtfdng phap chu dao la chieu bien thien ham so va cac Ta T i m gia t r i nho nha't cua bieu thiirc: P = 4 + 9 •+ - j|.phi/dng phap khac (thi du nhiC dtfa vao m o t ba't dang thuTc cho tru'dc, hoac suT s/ Idung bat ding thiJc Cosi, hoSc dufa vao cac kien thiJc ve hinh hoc giai tich,...) up Hitiing ddn gidi ICac b a i todn ay da m i n h hoa ro net cho nhflYig dieu ma chung t o i da dua ra ro X e m Icfi g i a i chi tiet trong bai 2, §2, chu^dng 1 cuon sach nay. j, /g B a i 6. (De thi tuyen sinh Dai hoc, Cao dang khoi D) om 'trong phan mc( dau cua §2, chufdng 4 n^y. Cho X , y la cac so thifc khong a m va thoa man dieu k i e n : x + y = 1. T i m gia Bai l l . C h o x > 0 , y > O v a x + y= 1. " .c t r i Idn nhat va nho nhat cua bieu thtfc: S - (ix^ + 3 y ) ( 4 y ^ + 3 x ) + 2 5 x y . T i m gia t r i be nha't cua bieu thi?c: P = ok x'+y-^ xy HUdng ddn gidi bo Hitdng dan gidi X e m Icfi g i a i chi tiet trong bai 6, §2, chtfcfng 1 cuon sdch nay. ce em IcJi g i a i c h i tiet trong phan nhan x e t ciia b a i 9, muc 1.2, §1, chiTdng 2 B a i 7. (De thi tuyen sinh Dai hoc, Cao dang khoi B) fa cuon sach n^y. Cho cac so thyc x, y thoa man dieu k i e n : (x + y)^ + 4xy > 2. T i m gia t r i nho w. 1 1 nha't cua b i e u thiJc: i giai v^n t^t sau: Di/a P ve dang: P = ww l-3yx xy A = 3(x'* + y ^ + x V ) - 2 ( x 2 + y 2 ) + l . HUdnjg ddn gidi at t = x y 0 < t < - va dufa v6 xet chieu bien thien cua ham so': 4J X e m IcJi g i a i chi tie't trong b a i 7, §2, chtfdng 1 cuon sach nay. 1 1 m\%.(De thi tuyen sinh Cao dang khoi A) f(t) = l-3t"^t • Cho X , y la hai so thifc thoa man: x S y^ = 2. Tiifd6c6:minP = 4+2>/3 . T i m gia t r i Idn nha't, nh6 nha't cua bieu thiJc: ^hqn xet: L d i giai la gon g^ng vk tif nhien! P = 2(x^ + y ^ ) - 3 x y . 221
Chuy6n dg BDHSG To^n g\A trj I6n nhfl't g\i tr| nh6 nhat - Phan Huy Kh5i Cty TNHH MTV DWH Khang Vigt Bai 12. Chi) x, y , /. l a b a s o thiTc diTdng s a o c h o x + y + z = 3. j^h^n xet: Ta c6 each giai khdc bang phifdng phap ba't dang thiJc sau day: Tim g i a tri Idn n h a t c i i a b i e u thuTc P = (x + y)(y + z)(z + x ) - ^ - ^ - s / z . A p dung bat d i n g thtfc Cosi ta c6 x^ + ^ + ^/x + ^ > 4^x^(^f = 4x hayx^+3^>4x. (1) A p d u n g h a n g d i l n g thuTc x^ + y^ + z' = (x + y + zf - 3(x = y)(y + z)(z + x ) , DSu b i n g trong (1) xay ra o X-'=\/x o x = 1 ( d o x > 0 ) . i g i a t h i c t X + y + z = 3, t a C O / - '! i Lap luan ti/dng M, ta c6 y^ + 3 ^ > 4 y , (2) x ' + y ' + z ' = 2 7 - 3 ( x + y)(y + z)(z + x). (1) / Tir ( 1 ) ta CO 3P = 3(x + y)(y + z)(z + X ) - 3 ( ^ + ^ + N/Z) z-''+3^>4z. (3) 01 I D S U b^ng trong (2) (3) tiftfng tirng xay ra o x = y = z = 1. oc = 27-(x-' + y ' + z ' ) - 3 ( ^ + ^ + ^) Cong turng ve (1) (2) (3) va c6 x ' + y ' + z^ + 3 + 3/^ + ^ ) > 12 (4) iH = 27 '(x-^+3^) +( y ^ + 3 ^ ) + ( 7 / + 3 ^ ) " Da I b k n g hi trong (4) xSy ra o x = y = z = 1. , , j , . , y^^j.,, ^ . ^. , , = 27- + 3 ^ - 4x) + ( y ' + 3 ^ - 4y) + ( z ' + 3 ^ - 4z)] - 1 2 (2) nT I A p dung h^ng d i n g thut ta Viet lai (4) diMi dang sau: uO (chu y do X + y + z = 3) (x + y + z ) ' - 3 ( x + y)(y + z)(z + x ) + 3 ( ^ + ^ +^ ) > 12 ie > • Xet ham so f(t) = t^ + 3 ^ - 4 t v d i 0 < t < 1. iL = > 2 7 - 3 ( x + y)(y + z)(z + x ) + 3 ( ^ + 3/5^ + ^ ) > 1 2 y:>numu . Ta ' Taco r(t)==3t2+^-4 , s/ •(x + y)(y + z)(z + x ) - ( ^ + 3 ^ + ^ ) 0
CtyTNHH MTV OWH Khang Vi^t Chuy6n dJ BDHSG To^n glA tr| Ifln nhat giA trj nh6 nhat - Phan Huy Khii Khid6lir{l)xelhamsf>F(t) = l + - ^ + 2 = l + - j ^ + 2 vc'li 1< t < V2 . ' >-'-^^'. (4, 2 Vi xy + yz + zx = 1, nen x, y, z khong the dong thdi b^ng 0. Taco F ' ( t ) - 1 ~ < 0 V l 6 (l;^/2 Sau khi cong lifng ve (2) (3) (4), ta c6 P > ^ (x^ + y^ + z \) (t-lr ^ min F(t) = F(N/2) = 4 + 3V2 o t = >/2. Da'u bSng trong (5) xay ra dong thcJi c6 da'u b^ng trong (2) (3) (4) / 01 ^ x =y=z = — . , •MM\(rjim:T- Tir (2) suy ra min F(x) = min F(t) = 4 + 3 V 2 < » t = N / 2 o x - - . oc 0 0 V 0 < X < 1. Khi do F(x) la h^m dong bien tren 0 < x < 1, nhif Nhir vay ta c6
Cty TNHH MTV DWH Khang Vijt Chuyfin di BDHSG Toan gia trj lOn nhait va gia trj nh6 nhift - Phan Huy KhSi \2 2. Neu F'(x) < O V O < x < l . I +1 t^ +1 t^ +2t + l J = f'(t), Thay (3) vao (2) va c6 P = —+• Khi do F(x) la ham nghjch bien tren 0 < x < 1. Tir do vdi moi 0 < x < 1, ta c6 t^-t +1 t^-t + 1 ^t^-t +1 P W , F(0) = ^ . - 1 - . (1 ^ yX • - z ) • ; " " " t^ +2t + l ^ vdi f(t) = vdi t e z+l y +1 1 + y + z + yz :? v," t^-t + 1 Do y V < yz va y, z deu 6 [0; 1] F(x) < 1 V X G [0; 1]. ^m^^ im Hif.> -, 41 >H *• • 3. Neu F'(x) CO dau thay doi tren [0; 1]. Do F'(x) la ham dong bien tren [0; Ta CO f'(t) = va c6 bang bien thien sau (t^-t + l)' / 01 nSn C h i CO the la t -1 1 oc 0 i; 1 ..".^v^... f'(t) 0 + 0 iH F'(x) - 0 + f(t) Da F(x) hi . t==1i oo xx = yv := i-' nT Do F(0) < 1; F ( l ) < 1, nen suy ra vdi moi 0 < x < 1, ta luon c6 F(x) < 1. Taco maxf(t) = 4 , n e n m a x P = 16c:>t (?:>sfun6v; 2 uO teR NhU"vay ket hcJp lai, ta luon c6 P < 1. Vay max? = 1. jj,^,^. .^^^^ Nhgn xet: Xem each giai bai toan tren bang phiTdng phap bat dang thuTc trong bai ie Nhqn xet: 11, §2, chiTdng 1 cuon sach nay. < iL 1. Trong bai toan tren mSc dau bieu thffc P phii thupc vao ba bien x. y, z nhifng ta i'f •^:£.mk~'i> 0 > < t } . + (|jdi "tot Ta da coi no chi la ham cua mot bien (bien x chang han). Sau do sur dung phtfdng s/ phap chieu bien thien ham so de giai bai toan da cho. ." '' • G&m^ rauiNGnuiPMifiiGUiiiiiiiiyrisf7 up TlM GUI TR| UlN NH/fr. NHi NHfr Gift HiUH Stf 2. CAch giai nay cung c6 the ap dung dc giai dc thi tuyen sinh Cao d^ng dai hoc ro khoiA-2011:Chox,y,ze [l;41saochox>y;x>z. , /g y z Gia sur ta phai tim maxf(x) hoac m i n f ( x ) , cf day D la mien xac dinh cua om Tim gid tri nho nhaft cua bieu thiJc P = 2x + 3y y+z z+x X€D X6D .c (xem Icfi giai trong bai 1, §2, chiTrtng 1 cuon sach n^y!) bien so x. * " ' • " ' ' ok Khi do de suT dung phiTcfng phap mien gia trj h^m so de giai bai toan tren ta » a i 16. (De thi Tuyen sinh Dai hoc Cao ddn^ khdi A) bo lamnhU'sau: : o :•*••;
Chuy6n ai BDHSG ToAn gi^ trj Mn nhait vk g\i trj nh6 nhSt - Phan Huy Kh5i Cty TNHH MTV DWH Khang Vi^t Bai 1. Tim gia trj Idn nhal nho nhat cua ham so: .. 5 3 niaxf(x)= — x = 2; min f(x) = - x = - 4 ^ 2 x ^ + 7 x + 23 ^ xeR 2 xeR 2 Ta thu lai ket qua Iren. Ban ihich each giai nao?. si/J s v , « x^ +2X + 10 2 Ta lai c6 the giai bang phi/cfng phap bat dang thtfc nhiT sau: HuAng dan gidi Gpi m m gia Iri tuy y ciia f(x), khi do phiTclng Irinh sau day (an x) f ( , ) . ^ ^ t t 2 ^ . 2 . - ^ ^ i l l _ = 2 + 3. (*) x ^ + 2 x + 10 x ^ + 2 x + 10 (x + l ) ^ + 9 ' 2x^ +7x + 2 3 _ x+1 x+1 x+1 / : • x ^ + 2 x + 10 Ta c6: 01 CO nghiem. V i x^ + 2x + 10 > 0 (Vx), nen I (x + 1)^ + 9 (X + 1)^ + 9 6 X +1 6 oc (l)2x' + 7x + 23 = m(x^ + 2 x + 1 0 ) (Theo bat ding thtfc Cosi, ta c6: (x + 1)' + 9 > 6|x +1|) iH o ( m - 2 ) x ^ + (2m-7)x+10m-23 = 0 (2) =1 Da => —