B\u00e0i 68 (trang 61 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b>\u00a0<\/span><\/p>\n

Ch\u1ee9ng minh c\u00e1c b\u1ea5t \u0111\u1eb3ng th\u1ee9c sau:<\/p>\n

<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) X\u00e9t h\u00e0m s\u1ed1 f(x) = tan x \u2013 x<\/p>\n

<\/p>\n

do \u0111\u00f3 h\u00e0m s\u1ed1 f(x) \u0111\u1ed3ng bi\u1ebfn tr\u00ean (0;\u03c0\/2)<\/p>\n

N\u00ean f(x) l\u00e0 h\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng (0;\u03c0\/2)<\/p>\n

V\u00ec f(0) = 0, n\u00ean khi x > 0 th\u00ec f(x) > f(0), t\u1ee9c l\u00e0 tan x \u2013 x > 0 hay tan x > x.<\/p>\n

b) X\u00e9t h\u00e0m s\u1ed1<\/p>\n

<\/p>\n

=tan^{2<\/sup>\u2061x-x2<\/sup>\u2061=(tan\u2061x-x)(tan\u2061x+x)>0 v\u1edbi m\u1ecdi x \u2208(0;\u03c0\/2) v\u00e0 do c\u00e2u a.<\/p>\n}

N\u00ean f(x) l\u00e0 h\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng (0;\u03c0\/2)<\/p>\n

V\u00ec f(0) = 0 n\u00ean khi x > 0 th\u00ec f(x) > f(), t\u1ee9c l\u00e0 tan\u2061x-x-x^3\/3>0 hay tan\u2061x>x+x^3\/3 v\u1edbi x \u2208(0;\u03c0\/2)<\/p>\n

B\u00e0i 69 (trang 61 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

X\u00e9t chi\u1ec1u bi\u1ebfn thi\u00ean v\u00e0 t\u00ecm c\u1ef1c tr\u1ecb (n\u1ebfu c\u00f3) c\u1ee7a h\u00e0m s\u1ed1 sau:<\/p>\n

<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

<\/p>\n

n\u00ean h\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng (-1\/3; +\u221e). Suy ra h\u00e0m s\u1ed1 ch\u1ec9 c\u00f3 c\u1ef1c ti\u1ec3u t\u1ea1i x=-1\/3 v\u00e0 gi\u00e1 tr\u1ecb c\u1ef1c ti\u1ec3u b\u1eb3ng y(-1\/3)=0.<\/p>\n

b) T\u1eadp x\u00e1c \u0111\u1ecbnh D = [0; 4],<\/p>\n

<\/p>\n

=> H\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng (0; 2), ngh\u1ecbch bi\u1ebfn tr\u00ean kho\u1ea3ng (2; 4). H\u00e0m s\u1ed1 ch\u1ec9 c\u00f3 m\u1ed9t gi\u00e1 tr\u1ecb c\u1ef1c \u0111\u1ea1i t\u1ea1i x = 2 v\u00e0 b\u1eb1ng 2.<\/p>\n

c) y=x+\u221ax. TX\u0110: D = [0; +\u221e)<\/p>\n

<\/p>\n

V\u1eady h\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng (0; +\u221e)<\/p>\n

Suy ra h\u00e0m s\u1ed1 c\u00f3 m\u1ed9t gi\u00e1 tr\u1ecb c\u1ef1c ti\u1ec3u t\u1ea1i x = 0 v\u00e0 gi\u00e1 tr\u1ecb c\u1ef1c ti\u1ec3u b\u1eb1ng y(0) = 0<\/p>\n

d) y=x-\u221ax, TX\u0110: D = [0; +\u221e)<\/p>\n

<\/p>\n

H\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng (1\/4; +\u221e), ngh\u1ecbch bi\u1ebfn tr\u00ean kho\u1ea3ng (0;1\/4).<\/p>\n

V\u1eady h\u00e0m s\u1ed1 c\u00f3 m\u1ed9t gi\u00e1 tr\u1ecb c\u1ee9c ti\u1ec3u t\u1ea1i x=1\/4<\/p>\n

V\u00e0 gi\u00e1 tr\u1ecb c\u1ef1c ti\u1ec3u b\u1eb1ng y(1\/4)=-1\/4<\/p>\n

B\u00e0i 70 (trang 61 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b>\u00a0<\/span><\/p>\n

Ng\u01b0\u1eddi ta \u0111\u1ecbnh l\u00e0m m\u1ed9t c\u00e1i h\u1ed9p kim lo\u1ea1i h\u00ecnh tr\u1ee5 c\u00f3 th\u1ec3 t\u00edch V cho tr\u01b0\u1edbc. t\u00ecm b\u00e1n k\u00ednh \u0111\u00e1y r v\u00e0 chi\u1ec1u cao h c\u1ee7a h\u00ecnh tr\u1ee5 sao cho \u00edt t\u1ed1n kim lo\u1ea1i nh\u1ea5t<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

<\/p>\n

G\u1ecdi b\u00e1n k\u00ednh c\u1ee7a h\u00ecnh tr\u1ee5 l\u00e0 a, 0 < x < V<\/p>\n

=> Chi\u1ec1u cao c\u1ee7a h\u00ecnh tr\u1ee5 l\u00e0<\/p>\n

<\/p>\n

\u0110\u1ec3 l\u00e0m m\u1ed9t c\u00e1i h\u1ed9p kim lo\u1ea1i h\u00ecnh tr\u1ee5 \u00edt t\u1ed1n kim lo\u1ea1i nh\u1ea5t th\u00ec di\u1ec7n t\u00edch to\u00e0n ph\u1ea7n c\u1ee7a h\u00ecnh tr\u1ee5 b\u00e9 nh\u1ea5t.<\/p>\n

<\/p>\n

B\u00e0i 71 (trang 62 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

Chu vi m\u1ed9t tam gi\u00e1c l\u00e0 16 cm, \u0111\u1ed9 d\u00e0i m\u1ed9t c\u1ea1nh tam gi\u00e1c l\u00e0 6 cm. t\u00ecm \u0111\u1ed9 d\u00e0i hai c\u1ea1nh c\u00f2n l\u1ea1i c\u1ee7a tam gi\u00e1c sao cho tam gi\u00e1c c\u00f3 di\u1ec7n t\u00edch l\u1edbn nh\u1ea5t.<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

G\u1ecdi \u0111\u1ed9 d\u00e0i ba c\u1ea1nh c\u1ee7a tam gi\u00e1c l\u00e0 6, x v\u00e0 10 \u2013 x, v\u1edbi 0 < x < 10<\/p>\n

\u00c1p d\u1ee5ng c\u00f4ng th\u1ee9c H\u00ea-r\u00f4ng ta c\u00f3 di\u1ec7n t\u00edch tam gi\u00e1c l\u00e0:<\/p>\n

<\/p>\n

\u0110\u1ec3 di\u1ec7n t\u00edch tam gi\u00e1c l\u1edbn nh\u1ea5t th\u00ec h\u00e0m s\u1ed1 y=-x^{2<\/sup>+10x-16 \u0111\u1ea1t gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t tr\u00ean (0; 10).<\/p>\n}

Ta c\u00f3 y’=-2x+10;y’=0 <=> x = 5<\/p>\n

H\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean (0; 5), ngh\u1ecbch bi\u00ean tr\u00ean kho\u1ea3ng (5; 10)<\/p>\n

V\u1eady h\u00e0m s\u1ed1 \u0111\u1ea1t gi\u00e1 tr\u1ecb l\u1edbn nh\u1ea5t t\u1ea1i x = 5.<\/p>\n

Khi \u0111\u00f3 di\u1ec7n t\u00edch c\u1ee7a tam gi\u00e1c l\u1edbn nh\u1ea5t b\u1eb1ng 6 \u221a2.<\/p>\n

V\u1eady \u0111\u1ed3 d\u00e0i c\u1ea1nh c\u00f2n l\u1ea1i \u0111\u1ec1u l\u00e0 5 cm.<\/p>\n

B\u00e0i 72 (trang 62 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

Cho h\u00e0m s\u1ed1<\/p>\n

<\/p>\n

a) Kh\u1ea3o s\u00e1t s\u1ef1 bi\u1ebfn thi\u00ean v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 \u0111\u00e3 cho.<\/p>\n

b) Ch\u1ee9ng minh r\u1eb1ng Ph\u01b0\u01a1ng tr\u00ecnh f(x) = 0 c\u00f3 ba nghi\u1ec7m ph\u00e2n bi\u1ec7t.<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) TX\u0110: R<\/p>\n

y’=x^{2<\/sup>-4x=x(x-4);y’=0 <=> x = 0; x = -4<\/p>\n}

H\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean c\u00e1c kho\u1ea3ng (-\u221e,0)v\u00e0 (4; +\u221e)<\/p>\n

H\u00e0m s\u1ed1 ngh\u1ecbch bi\u1ebfn tr\u00ean kho\u1ea3ng (0; 4)<\/p>\n

H\u00e0m s\u1ed1 \u0111\u1ea1t c\u1ef1c \u0111\u1ea1i t\u1ea1i x = 0; y_{C\u0110<\/sub>=y(0)=17\/3<\/p>\n}

H\u00e0m s\u1ed1 \u0111\u1ea1t c\u1ef1c ti\u1ec3u t\u1ea1i x = 4, y_{CT<\/sub>=y(4)=-5<\/p>\n}

<\/p>\n

y”=2x-4; y”=0 <=> x = 2<\/p>\n

B\u1ea3ng x\u00e9t d\u1ea5u<\/p>\n

<\/p>\n

B\u1ea3ng bi\u1ebfn thi\u00ean<\/p>\n

<\/p>\n

\u0110\u1ed3 th\u1ecb<\/p>\n

<\/p>\n

b) \u0110\u1ed3 th\u1ecb h\u00e0m s\u1ed1<\/p>\n

<\/p>\n

c\u1eaft tr\u1ee5c ho\u00e0nh t\u1ea1i 3 \u0111i\u1ec3m ph\u00e2n bi\u1ec7t. v\u1eady ph\u01b0\u01a1ng tr\u00ecnh f(x) = 0 c\u00f3 ba nghi\u1ec7m ph\u00e2n bi\u1ec7t.<\/p>\n

B\u00e0i 73 (trang 62 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

Cho h\u00e0m s\u1ed1 f(x) = x^{3<\/sup>+px+q<\/p>\n}

a) T\u00ecm \u0111i\u1ec1u ki\u1ec7n \u0111\u1ed1i v\u1edbi p v\u00e0 q \u0111\u1ec3 h\u00e0m s\u1ed1 f c\u00f3 c\u1ef1c \u0111\u1ea1i v\u00e0 m\u1ed9t c\u1ef1c ti\u1ec3u.<\/p>\n

b) Ch\u1ee9ng minh r\u1eb1ng n\u1ebfu c\u00f3 gi\u00e1 tr\u1ecb c\u1ef1c \u0111\u1ea1i v\u00e0 gi\u00e1 tr\u1ecb c\u1ef1c ti\u1ec3u tr\u00e1i d\u1ea5u th\u00ec Ph\u01b0\u01a1ng tr\u00ecnh x^{3<\/sup>+px+q = (1) c\u00f3 3 nghi\u1ec7m ph\u00e2n bi\u1ec7t.<\/p>\n}

c) Ch\u1ee9ng minh r\u1eb1ng \u0111i\u1ec1u ki\u1ec7n c\u1ea7n v\u00e0 \u0111\u1ee7 \u0111\u1ec3 Ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 ba nghi\u1ec7m ph\u00e2n bi\u1ec7t l\u00e0 4p^{3<\/sup>+27q2<\/sup><0<\/p>\n}

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) f'(x)=3x^{2<\/sup>+p<\/p>\n}

\u0110\u1ec3 h\u00e0m s\u1ed1 f c\u00f3 m\u1ed9t c\u1ef1c \u0111\u1ea1i v\u00e0 m\u1ed9t c\u1ef1c ti\u1ec3u th\u00ec ph\u01b0\u01a1ng tr\u00ecnh f\u2019(x) = 2 c\u00f3 2 nghi\u1ec7m ph\u00e2n bi\u1ec7t v\u00e0 f\u2019(x) \u0111\u1ed1i d\u1ea5u qua c\u00e1c \u0111i\u1ec3m \u0111\u00f3. V\u1eadt p < 0<\/p>\n

b) C\u00e1ch 1.<\/p>\n

D\u1ea1ng \u0111\u1ed3 th\u1ecb nh\u01b0 h\u00ecnh v\u1ebd.<\/p>\n

<\/p>\n

\u0110\u1ed3 th\u1ecb c\u1eaft tr\u1ee5c ho\u00e0nh t\u1ea1i ba \u0111i\u1ec3m ph\u00e2n bi\u1ec7t n\u00ean ph\u01b0\u01a1ng tr\u00ecnh x^{3<\/sup>+px+q= 0 c\u00f3 3 nghi\u1ec7m ph\u00e2n bi\u1ec7t.<\/p>\n}

C\u00e1ch 2.<\/p>\n

H\u00e0m s\u1ed1 f(x) = x^{3<\/sup>+px+q li\u00ean t\u1ee5c tr\u00ean R v\u00e0 c\u00f3<\/p>\n}

<\/p>\n

f_{C\u0110<\/sub>=f(x1<\/sub>\u00a0),fCT<\/sub>=f(x2<\/sub>\u00a0)<\/p>\n}

<\/p>\n

n\u00ean t\u1ed3n t\u1ea1i s\u1ed1 a sao cho f(a) < 0, a<x_{1<\/sub><<\/p>\n}

V\u00ec f(a), f_{C\u0110<\/sub><0 n\u00ean ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 \u00edt nh\u1ea5t 1 nghi\u1ec7m thu\u1ed9c (a,x1<\/sub>)<\/p>\n}

f(x_{1<\/sub>\u00a0);f(x^{2<\/sup>\u00a0)<0 n\u00ean ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 \u00edt nh\u1ea5t 1 nghi\u1ec7m thu\u1ed9c (x1<\/sub>,x2<\/sub>)<\/p>\n}}

<\/p>\n

n\u00ean t\u1ed3n t\u1ea1i m\u1ed9t s\u1ed1 b > x_{2<\/sub>\u00a0sao cho f(b) > 0<\/p>\n}

V\u00ec f(x^{2<\/sup>\u00a0),f(b)<0 n\u00ean ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 \u00edt nh\u1ea5t 1 nghi\u1ec7m thu\u1ed9c (x2<\/sup>,b)<\/p>\n}

Do Ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc ba c\u00f3 nhi\u1ec1u nh\u1ea5t l\u00e0 3 nghi\u1ec7m. v\u1eady Ph\u01b0\u01a1ng tr\u00ecnh x^{3<\/sup>+px+q=0 c\u00f3 3 nghi\u1ec7m ph\u00e2n bi\u1ec7t.<\/p>\n}

Ch\u00fa \u00fd: kh\u1eb3ng \u0111\u1ecbnh tr\u00ean \u0111\u00fang v\u1edbi Ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc ba t\u1ed5ng qu\u00e1t.<\/p>\n

<\/p>\n

G\u1ecdi x_{1<\/sub>,x2<\/sub>\u00a0l\u00e0 hai \u0111i\u1ec3m c\u1ef1c tr\u1ecb c\u1ee7a h\u00e0m s\u1ed1.<\/p>\n}

Theo c\u00e2u b, ta c\u00f3 \u0111i\u1ec1u ki\u1ec7n c\u1ea7n v\u00e0 \u0111\u1ee7 \u0111\u1ec3 Ph\u01b0\u01a1ng tr\u00ecnh (1) c\u00f3 ba nghi\u1ec7m ph\u00e2n bi\u1ec7t l\u00e0 gi\u00e1 tr\u1ecb c\u1ef1c \u0111\u1ea1i v\u00e0 c\u1ef1c ti\u1ec3u tr\u00e1i d\u1ea5u nhau, ngh\u0129a l\u00e0 y_{CD<\/sub>.yCT<\/sub><0 <=> f(x1<\/sub>\u00a0).f(x2<\/sub>\u00a0)<0<\/p>\n}

<\/p>\n

B\u00e0i 74 (trang 62 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b>\u00a0<\/span><\/p>\n

Cho h\u00e0m s\u1ed1 f(x) = x^{3<\/sup>-3x+1<\/p>\n}

a) Kh\u1ea3o s\u00e1t s\u1ef1 bi\u1ebfn thi\u00ean v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1.<\/p>\n

b) Vi\u1ebft ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u1ed3 th\u1ecb t\u1ea1i \u0111i\u1ec3m u\u1ed1n U c\u1ee7a n\u00f3.<\/p>\n

c) G\u1ecdi (d_{m<\/sub>) l\u00e0 \u0111\u01b0\u1eddng th\u1eb3ng \u0111i qua U v\u00e0 c\u00f3 h\u1ec7 s\u1ed1 g\u00f3c m. t\u00ecm c\u00e1c gi\u00e1 tr\u1ecb c\u1ee7a m sao cho \u0111\u01b0\u1eddng th\u1eb3ng (dm<\/sub>) c\u1eaft \u0111\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 \u0111\u00e3 cho t\u1ea1i 3 \u0111i\u1ec3m ph\u00e2n bi\u1ec7t.<\/p>\n}

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) TX\u0110: D = R<\/p>\n

y’=3x^{2<\/sup>-3;y’=0 <=> 3x2<\/sup>-3=0 <=> 3(x2<\/sup>-1)=0 <=> x = 1; x = -1<\/p>\n}

H\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean c\u00e1c kho\u1ea3ng (-\u221e,-1)v\u00e0 (1; +\u221e), ngh\u1ecbch bi\u1ebfn tr\u00ean kho\u1ea3ng (1-; 1)<\/p>\n

y_{C\u0110<\/sub>=y(-1)=3;yCT<\/sub>=y(1)=-1<\/p>\n}

<\/p>\n

B\u1ea3ng x\u00e9t d\u1ea5u<\/p>\n

<\/p>\n

B\u1ea3ng bi\u1ebfn thi\u00ean<\/p>\n

<\/p>\n

\u0110\u1ed3 th\u1ecb<\/p>\n

<\/p>\n

b) Ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u1ed3 th\u1ecb t\u1ea1i \u0111i\u1ec3m u\u1ed1n U l\u00e0: y-1=f'(0)(x-0)<\/p>\n

<=> y-1=-3 ,=> y=-3x+1<\/p>\n

c) Ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01b0\u1eddng th\u1eb3ng (_{m<\/sub>) \u0111i qua \u0111i\u1ec3m u\u1ed1n U v\u00e0 c\u00f3 h\u1ec7 s\u1ed1 g\u00f3c m c\u00f3 d\u1ea1ng y=mx+1<\/p>\n}

Ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a \u0111\u01b0\u1eddng th\u1eb3ng (d_{m<\/sub>) v\u00e0 \u0111\u1ed3 th\u1ecb l\u00e0 nghi\u1ec7m c\u1ee7a Ph\u01b0\u01a1ng tr\u00ecnh x^{3<\/sup>-3x+1=mx+1<=> x3<\/sup>-(3+m)x=0 <=> x(2<\/sup>-m-3)=0 (*)<\/p>\n}}

\u0110\u1ec3 \u0111\u01b0\u1eddng th\u1eb3ng d_{m<\/sub>\u00a0c\u1eaft \u0111\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 \u0111\u00e3 cho t\u1ea1i ba \u0111i\u1ec3m ph\u00e2n bi\u1ec7t th\u00ec Ph\u01b0\u01a1ng tr\u00ecnh x^{3<\/sup>-m-3=0 c\u00f3 2 nghi\u1ec7m ph\u00e2n bi\u1ec7t \u2260 0.<\/p>\n}}

<=> m+3>0 ,=> m > -3<\/p>\n

V\u1eady v\u1edbi m > -3 l\u00e0 gi\u00e1 tr\u1ecb c\u1ea7n t\u00ecm.<\/p>\n

B\u00e0i 75 (trang 62 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b>\u00a0<\/span><\/p>\n

Cho h\u00e0m s\u1ed1 y=x^{4<\/sup>-(m+1) x2<\/sup>+m<\/p>\n}

a) Kh\u1ea3o s\u00e1t s\u1ef1 bi\u1ebfn thi\u00ean v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 m = 2<\/p>\n

b) T\u00ecm c\u00e1c gi\u00e1 tr\u1ecb c\u1ee7a m sao cho \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 c\u1eaft tr\u1ee5c ho\u00e0nh t\u1ea1i b\u1ed1n \u0111i\u1ec3m, t\u1ea1o th\u00e0nh 3 \u0111o\u1ea1n th\u1eb3ng c\u00f3 \u0111\u1ed9 d\u00e0i b\u1eb1ng nhau.<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) V\u1edbi m = 2. H\u00e0m s\u1ed1 y=x^{4<\/sup>-3x2<\/sup>+2<\/p>\n}

TX\u0110: D = R<\/p>\n

y’=4x^{3<\/sup>-5x=2x(2x2<\/sup>-3);y’=0 <=> x = 0; x=\u221a6\/2;x= -\u221a6\/2<\/p>\n}

<\/p>\n

\u0110i\u1ec3m c\u1ef1c \u0111\u1ea1i x = 0; y_{CD<\/sub>=y(0)=2<\/p>\n}

Gi\u1edbi h\u1ea1n<\/p>\n

<\/p>\n

B\u1ea3ng bi\u00ean thi\u00ean<\/p>\n

<\/p>\n

\u0110\u1ed3 th\u1ecb<\/p>\n

\u0110\u1ed3 th\u1ecb c\u1eaft tr\u1ee5c tung t\u1ea1i (0; 2)<\/p>\n

C\u1eaft tr\u1ee5c ho\u00e0nh t\u1ea1i 4 \u0111i\u1ec3m (-\u221a2;0);(-1;0);(1;0);(\u221a2,0)<\/p>\n

<\/p>\n

b) \u0110\u1eb7t t=x^{2<\/sup>, \u0111i\u1ec1u ki\u1ec7n t\u22650. Ho\u00e0nh \u0111\u1ed9 giao \u0111i\u1ec3m c\u1ee7a \u0111\u1ed3 th\u1ecb v\u00e0 tr\u1ee5c ho\u00e0nh l\u00e0 nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh.<\/p>\n}

x^{4<\/sup>-(m+1) x2<\/sup>+m=0 (1)<\/p>\n}

<=> t^{2<\/sup>-(m+1)t+m=0 (2)<\/p>\n}

\u0110\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 c\u1eaft tr\u1ee5c tung t\u1ea1i b\u1ed1n \u0111i\u1ec3m t\u1ea1o th\u00e0nh 3 \u0111o\u1ea1n th\u1eb3ng c\u00f3 \u0111\u1ed9 dai b\u1eb1ng nhau, t\u1ee9c 4 \u0111i\u1ec3m c\u00f3 ho\u00e0nh \u0111\u1ed9 l\u1eadp th\u00e0nh c\u1ea5p s\u1ed1 c\u1ed9ng.<\/p>\n

<=> Ph\u01b0\u01a1ng tr\u00ecnh (2) c\u00f3 2 nghi\u1ec7m d\u01b0\u01a1ng t_{1<\/sub>,t2<\/sub>\u00a0(v\u1edbi t1<\/sub>\u00a0< t2<\/sub>) th\u00f5a m\u00e3n \u0111i\u1ec1u ki\u1ec7n:<\/p>\n}

\u221a(t_{2<\/sub>\u00a0)-\u221a(t1<\/sub>\u00a0)=\u221a(t1<\/sub>\u00a0)-(-\u221a(t1<\/sub>\u00a0))<=> \u221a(t2<\/sub>\u00a0)=3 \u221a(t1<\/sub>\u00a0) <=> t2<\/sub>=9t1<\/sub><\/p>\n}

Ph\u01b0\u01a1ng tr\u00ecnh (2) c\u00f3 2 nghi\u1ec7m kh\u00e1c nhau l\u00e0:<\/p>\n

<\/p>\n

V\u1eady v\u1edbi m = 9 ho\u1eb7c m=1\/9 th\u00ec \u0111\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 c\u1eaft tr\u1ee5c ho\u00e0nh t\u1ea1i 4 \u0111i\u1ec3m, t\u1ea1o th\u00e0nh 3 \u0111o\u1ea1n th\u1eb3ng b\u1eb1ng nhau.<\/p>\n

\n

\n

B\u00e0i 76 (trang 63 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b>\u00a0<\/span><\/p>\n

Cho h\u00e0m s\u1ed1 f(x)=x^{4<\/sup>-x_{2<\/sub><\/p>\n}}

a) Kh\u1ea3o s\u00e1t s\u1ef1 bi\u1ebfn thi\u00ean v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 \u0111\u00e3 cho.<\/p>\n

b) T\u1eeb \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 f(x) suy ra \u0111\u1ed3 th\u1ecb c\u1ee7a h\u00e0m s\u1ed1 y = |f(x)|<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) TX\u0110: D = R<\/p>\n

y’=4x^{3<\/sup>-2x=2x(2x2<\/sup>-1)=0 <=> x = 0; x=\u00b1\u221a2\/2<\/p>\n}

<\/p>\n

\u0110i\u1ec3m c\u1ef1c \u0111\u1ea1i x = 0; y_{C\u0110<\/sub>=y(0)=0<\/p>\n}

Gi\u1edbi h\u1ea1n:<\/p>\n<\/div>\n

<\/div>\n

\n

<\/p>\n

B\u1ea3ng bi\u00ean thi\u00ean<\/p>\n

<\/p>\n

\u0110\u1ed3 th\u1ecb<\/p>\n

\u0110i qua (0; 0); (-1; 0) v\u00e0 (1; 0)<\/p>\n

<\/p>\n

do \u0111\u00f3 \u0111\u1ed3 th\u1ecb y = |f(x)| g\u1ed3m.<\/p>\n

– Ph\u1ea7n \u0111\u1ed3 th\u1ecb tr\u00ean Ox c\u1ee7a \u0111\u1ed3 th\u1ecb (\u0111\u00e3 v\u1ebd)<\/p>\n

– Ph\u1ea7n \u0111\u1ed1i x\u1ee9ng ph\u1ea7n \u0111\u1ed3 th\u1ecb ph\u00eda d\u01b0\u1edbi Ox c\u1ee7a \u0111\u1ed3 th\u1ecb v\u00e0 v\u1ebd qua Ox.<\/p>\n

– \u0110\u1ed3 th\u1ecb (\u0111\u01b0\u1eddng n\u00e9t li\u1ec1n)<\/p>\n

<\/div>\n

\n

B\u00e0i 77 (trang 63 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b>\u00a0<\/span><\/p>\n

Cho h\u00e0m s\u1ed1<\/p>\n

<\/p>\n

a) Kh\u1ea3o s\u00e1t s\u1ef1 bi\u1ebfn thi\u00ean v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 v\u1edbi m = 1<\/p>\n

b) Ch\u1ee9ng minnh r\u1eb1ng v\u1edbi m\u1ecdi m \u2260 1\/2, c\u00e1c \u0111\u01b0\u1eddng cong (H_{m<\/sub>) \u0111\u1ec1u \u0111i qau hai \u0111i\u1ec3m c\u1ed1 \u0111\u1ecbnh A, B.<\/p>\n}

c) Ch\u1ee9ng minh r\u1eb1ng t\u00edch c\u00e1c h\u1ec7 s\u1ed1 g\u00f3c c\u1ee7a ti\u1ebfp tuy\u1ebfn v\u1edbi (H_{m<\/sub>) t\u1ea1i hai \u0111i\u1ec3m A v\u00e0 B l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 khi m bi\u1ebfn thi\u00ean.<\/p>\n}

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

<\/p>\n

N\u00ean h\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean c\u00e1c kho\u1ea3ng (-\u221e;1) v\u00e0 (1; +\u221e)<\/p>\n

<\/p>\n

Do \u0111\u00f3 \u0111\u01b0\u1eddng th\u1eb3ng x = 1 l\u00e0 ti\u1ec7m c\u1eadn \u0111\u1ee9ng.<\/p>\n

<\/p>\n

=> \u0111\u01b0\u1eddng th\u1eb3ng y=1\/2 l\u00e0 ti\u1ec7m c\u1eadn ngang.<\/p>\n

B\u1ea3ng bi\u1ebfn thi\u00ean<\/p>\n

<\/p>\n

\u0110\u1ed3 th\u1ecb<\/p>\n

C\u1eb7t tr\u1ee5c tung t\u1ea1i (0; 2)<\/p>\n

C\u1eb7t tr\u1ee5c ho\u00e0nh t\u1ea1i (4; 0)<\/p>\n

<\/p>\n

\u0110k: mx \u2260 1<\/p>\n

b) G\u1ecdi A(x, y) l\u00e0 \u0111i\u1ec3m c\u1ed1 \u0111\u1ecbnh c\u1ee7a \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 khi m thay \u0111\u1ed5i.<\/p>\n

Khi \u0111\u00f3 t\u1ecda \u0111\u1ed9 c\u1ee7a A th\u00f5a m\u00e3n Ph\u01b0\u01a1ng tr\u00ecnh sau \u2200m:<\/p>\n

<\/p>\n

=> y=\u00b11<\/p>\n

+ y = 1 => x = -2<\/p>\n

+ y = -1 => x = 2<\/p>\n

V\u1eady \u2200m \u2260 \u00b11\/2 \u0111\u1ed3 th\u1ecb h\u00e0m s\u1ed1 lu\u00f4n \u0111i qua hai \u0111i\u1ec3m c\u1ed1 \u0111\u1ecbnh A(-2; 1), B(2; -1)<\/p>\n

<\/p>\n

H\u1ec7 s\u1ed1 c\u1ee7a ti\u1ebfp tuy\u1ebfn v\u1edbi (H_{m<\/sub>) t\u1ea1i \u0111i\u1ec3m A l\u00e0 y\u2019(2), t\u1ea1i \u0111i\u1ec3m B l\u00e0 y\u2019(2)<\/p>\n}

Ta c\u00f3 :<\/p>\n

<\/div>\n

<\/div>\n

\n

B\u00e0i 78 (trang 63 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b><\/span><\/p>\n

a) V\u1ebd \u0111\u1ed3 th\u1ecb (T) c\u1ee7a h\u00e0m s\u1ed1 y=x^2-x+1 v\u00e0 \u0111\u1ed3 th\u1ecb (H) c\u1ee7a h\u00e0m s\u1ed1<\/p>\n

<\/p>\n

b) T\u00ecm giao \u0111i\u1ec3m c\u1ee7a hai \u0111\u01b0\u1eddng cong (T) v\u00e0 (H). ch\u1ee9ng minh r\u1eb1ng hai \u0111\u01b0\u1eddng cong \u0111\u00f3 c\u00f3 ti\u1ebfp tuy\u1ebfn chung t\u1ea1i giao \u0111i\u1ec3m chung c\u1ee7a ch\u00fang.<\/p>\n

c) X\u00e1c \u0111\u1ecbnh c\u00e1c kho\u1ea3ng tr\u00ean \u0111\u00f3 (H) n\u1eb1m ph\u00eda tr\u00ean ho\u1eb7c d\u01b0\u1edbi (H).<\/p>\n

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) \u0110\u1ed3 th\u1ecb<\/p>\n

<\/p>\n

b) D\u1ef1a v\u00e0o \u0111\u1ed3 th\u1ecb ta c\u00f3 giao \u0111i\u1ec3m c\u1ee7a (T) v\u00e0 (H) l\u00e0 A(0; 1)<\/p>\n

c) D\u1ef1a v\u00e0o \u0111\u1ed3 th\u1ecb ta th\u1ea5y tr\u00ean (-\u221e; -1) v\u00e0 (0; +\u221e), (T) n\u1eb1m tr\u00ean (H), tr\u00ean (-1; 0) (T) n\u1eb1m ph\u00eda d\u01b0\u1edbi (H).<\/p>\n

B\u00e0i 79 (trang 63 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):<\/b>\u00a0<\/span><\/p>\n

Cho h\u00e0m s\u1ed1<\/p>\n

<\/p>\n

a) Kh\u1ea3o s\u00e1t v\u00e0 v\u1ebd \u0111\u1ed3 th\u1ecb (C) c\u1ee7a h\u00e0m s\u1ed1.<\/p>\n

b) Ti\u1ebfp tuy\u1ebfn c\u1ee7a \u0111\u01b0\u1eddng cong (C) t\u1ea1i M(x_{0<\/sub>,y0<\/sub>) c\u1eaft ti\u1ec7m c\u1eadn \u0111\u1ee9ng v\u00e0 ti\u1ec7m c\u1eadn xi\u00ean t\u1ea1i hai \u0111i\u1ec3m A, B. ch\u1ee9ng minh r\u1eb1ng M l\u00e0 trung \u0111i\u1ec3m tr\u00ean \u0111\u01b0\u1eddng cong (C).<\/p>\n}

L\u1eddi gi\u1ea3i:<\/b><\/p>\n

a) TX\u0110: D = R \\ {0}<\/p>\n

<\/p>\n

H\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean c\u00e1c kho\u1ea3ng (-\u221e;-1)v\u00e0 (1; +\u221e), ngh\u1ecbch bi\u1ebfn tr\u00ean (-1; 0) v\u00e0 (0; 1)<\/p>\n

H\u00e0m s\u1ed1 \u0111\u1ea1t c\u1ef1c \u0111\u1ea1i t\u1ea1i x = -1 v\u00e0 y_{C\u0110<\/sub>=y(-1)=-2<\/p>\n}

H\u00e0m s\u1ed1 \u0111\u1ea1t c\u1ef1c ti\u1ec3u t\u1ea1i x = 1 v\u00e0 y_{CT<\/sub>=y(1)=2<\/p>\n}

Gi\u1edbi h\u1ea1n:<\/p>\n

<\/p>\n

V\u1eady \u0111\u01b0\u1eddng th\u1eb3ng x = 0 l\u00e0 ti\u1ec7m c\u1eadn \u0111\u1ee9ng.<\/p>\n

<\/p>\n

n\u00ean \u0111\u01b0\u1eddng th\u1eb3ng y = x l\u00e0 ti\u1ec7m c\u1eadn xi\u00ean.<\/p>\n

B\u1ea3ng bi\u1ebfn thi\u00ean<\/p>\n

<\/p>\n

\u0110\u1ed3 th\u1ecb<\/p>\n

<\/p>\n

b) Ph\u01b0\u01a1ng tr\u00ecnh ti\u1ebfp tuy\u1ebfn t\u1ea1i \u0111i\u1ec3m M(x_{0<\/sub>;y0<\/sub>) thu\u1ed9c (C) l\u00e0:<\/p>\n}

<\/p>\n

\u0394 c\u1eaft ti\u1ec7m c\u1eadn \u0111\u1ee9ng t\u1ea1i A.<\/p>\n

=> T\u1ecda \u0111\u1ed9 A(0;2\/x_{0<\/sub>\u00a0)<\/p>\n}

\u0394 c\u1eaft ti\u1ec7m c\u1eadn xi\u00ean t\u1ea1i B.<\/p>\n

=> T\u1ecda \u0111\u1ed9 B(2x_{0<\/sub>,2x0<\/sub>)<\/p>\n}

+ T\u1ecda \u0111\u1ed9 trung \u0111i\u1ec3m c\u1ee7a AB l\u00e0:<\/p>\n

<\/p>\n

V\u1eady M l\u00e0 trung \u0111i\u1ec3m c\u1ee7a \u0111o\u1ea1n AB.<\/p>\n

+ Kho\u1ea3ng c\u00e1ch t\u1eeb B \u0111\u1ebfn tr\u1ee5c Oy b\u1eb1ng 2x_{0<\/sub>\u00a0l\u00e0 \u0111\u1ed9 d\u00e0i \u0111\u01b0\u1eddng cao k\u1ebb t\u1eeb B c\u1ee7a OAB, OA c\u00f3 \u0111\u1ed9 d\u00e0i b\u1eb1ng 2\/x0<\/sub>\u00a0.<\/p>\n}

V\u1eady di\u1ec7n t\u00edch tam gi\u00e1c OAB l\u00e0 (1\/2).2x_{0<\/sub>.(2\/x0<\/sub>) =2 kh\u00f4ng \u0111\u1ed5i (kh\u00f4ng ph\u1ee5 thu\u1ed9c v\u00e0o v\u1ecb tr\u1ecb c\u1ee7a M \u2208(C).<\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"}

B\u00e0i 68 (trang 61 sgk Gi\u1ea3i T\u00edch 12 n\u00e2ng cao):\u00a0 Ch\u1ee9ng minh c\u00e1c b\u1ea5t \u0111\u1eb3ng th\u1ee9c sau: L\u1eddi gi\u1ea3i: a) X\u00e9t h\u00e0m s\u1ed1 f(x) = tan x \u2013 x do \u0111\u00f3 h\u00e0m s\u1ed1 f(x) \u0111\u1ed3ng bi\u1ebfn tr\u00ean (0;\u03c0\/2) N\u00ean f(x) l\u00e0 h\u00e0m s\u1ed1 \u0111\u1ed3ng bi\u1ebfn tr\u00ean kho\u1ea3ng (0;\u03c0\/2) V\u00ec f(0) = 0, n\u00ean khi x […]<\/p>\n","protected":false},"author":3,"featured_media":23598,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mi_skip_tracking":false,"tdm_status":"","tdm_grid_status":""},"categories":[1302],"tags":[1392,1393],"yoast_head":"\n