Lucas_meomeo_287
16-05-2009, 09:50 PM
PHƯƠNG PHÁP THAM SỐ PHỤ Áp dụng cho BDT Côsi
Ví dụ 1 : Cho x,y >= 0 thỏa mãn [Only registered and activated users can see links] 01
Tìm GTLN của biểu thức :
[Only registered and activated users can see links] 02%5Csqrt%7By%7D
Giải :
Đặt [Only registered and activated users can see links] 0a%20%5C%5C%7B%20y%20=%20b%7D
Áp dụng BDT Côsi cho 6 số :
[Only registered and activated users can see links] 3%20+%20a%5E3%20+%20a%5E3%20+%20a%5E3%20%5Cgeq%206 %5Csqrt%5B6%5D%7Bx%5E3.a%5E%7B15%7D%7D
[Only registered and activated users can see links] 3%20+%20b%5E3%20+%20b%5E3%20+%20b%5E3%20%5Cgeq%206 %5Csqrt%5B6%5D%7By%5E3.b%5E%7B15%7D%7D
Cộng vế theo vế :
[Only registered and activated users can see links] a%5E3%20+%20b%5E3%29%20%5Cgeq%206%28%5Csqrt%5B6%5D %7Ba%5E%7B15%7D%7D%5Csqrt%7Bx%7D%20+%20%5Csqrt%5B6 %5D%7Bb%5E%7B15%7D%7D.%5Csqrt%7By%7D%29%20=%206%28 %5Csqrt%7Ba%5E%7B5%7D%7D%5Csqrt%7Bx%7D%20+%20%5Csq rt%7Bb%5E%7B5%7D%7D.%5Csqrt%7By%7D%29%20%284%29
Vậy ta cần xác định a,b thỏa hệ :
[Only registered and activated users can see links] 0+%20b%5E3%20=%201%20%281%29%20%5C%5C%20%7B%5Csqrt %7Bb%5E5%7D%20=%202%5Csqrt%7Ba%5E5%7D%282%29%7D
Từ (2) :[Only registered and activated users can see links] B4%7D
thay vào (1) :
[Only registered and activated users can see links] %5D%7B64%7D%20=%201
[Only registered and activated users can see links] rac%7B1%7D%7B%5Csqrt%5B3%5D%7B1%20+%20%5Csqrt%5B5% 5D%7B64%7D%7D%7D
[Only registered and activated users can see links] 5Cfrac%7B%5Csqrt%5B5%5D%7B4%7D%7D%7B%5Csqrt%5B3%5D %7B1%20+%5Csqrt%5B5%5D%7B64%7D%20%7D%7D
Thay vào (4) :
[Only registered and activated users can see links] E3%20+%205%28a%5E3%20+%20b%5E3%29%20%5Cgeq%206%5Cs qrt%5B6%5D%7B%5Cfrac%7B1%7D%7B%281%20+%20%5Csqrt%5 B5%5D%7B64%7D%29%5E5%7D%7D%28%20%5Csqrt%7Bx%7D%20+ %202%5Csqrt%7By%7D%29
[Only registered and activated users can see links] 0%5Csqrt%5B6%5D%7B%5Cfrac%7B1%7D%7B%281%20+%20%5Cs qrt%5B5%5D%7B64%7D%29%5E5%7D%7D.P
[Only registered and activated users can see links] 0%5Csqrt%5B6%5D%7B%281%20+%20%5Csqrt%5B5%5D%7B64%7 D%29%5E5%7D
Ví dụ 2 : Tìm GTNN của hàm số :
[Only registered and activated users can see links] 5Csqrt%7B3%7D%7D%20-%20%5Csqrt%7Bx%28x%20-%201%29%7D với [Only registered and activated users can see links]
Giải :
Đặt [Only registered and activated users can see links] 20-%20a%7D
Áp dụng BDT Côsi :
[Only registered and activated users can see links] a%7D%7D.2%5Csqrt%7Bax%28x%20-%201%29%7D%20%5Cleq%20%5Cfrac%7B1%7D%7B2%5Csqrt%7B a%7D%7D.%28ax%20+%20x%20-%201%29
Ta xác định a sao cho :
[Only registered and activated users can see links] a%7D%7D%28a%20+%201%29%20=%20%5Cfrac%7B7%7D%7B4%5C sqrt%7B3%7D%7D
[Only registered and activated users can see links] 0+%201%29%5E2%20=%2049a
[Only registered and activated users can see links] 20-%2025a%20+%2012%20=%200
[Only registered and activated users can see links] %5Cfrac%7B3%7D%7B4%7D (vì [Only registered and activated users can see links])
Vậy :
[Only registered and activated users can see links] 7B%5Csqrt%7B3%7D%7D
[Only registered and activated users can see links] %7B%5Csqrt%7B3%7D%7D
Xảy ra [Only registered and activated users can see links] 4
Ví dụ 3 : Tìm GTLN của hàm số :
[Only registered and activated users can see links] +%20x%5E4%7D%20+%2013x%5E2.%5Csqrt%7B1%20-%20x%5E4%7D
Giải :
Đặt [Only registered and activated users can see links]
[Only registered and activated users can see links]
Áp dụng BDT Côsi :
[Only registered and activated users can see links] a%7D%7D2%5Csqrt%7Ba%7Dx%5E2%5Csqrt%7B1%20+%20x%5E4 %7D%20%5Cleq%20%5Cfrac%7B9%7D%7B2%5Csqrt%7Ba%7D%7D %28ax%5E4%20+%201%20+%20x%5E4%29
[Only registered and activated users can see links] Ba%20-%202%7D%7D2%5Csqrt%7Ba%20-%202%7Dx%5E2%5Csqrt%7B1%20-%20x%5E4%7D%20%5Cleq%20%5Cfrac%7B13%7D%7B2%5Csqrt% 7Ba%20-%202%7D%7D%5B%28a%20-%202%29x%5E4%20+%201%20-%20x%5E4%5D
Ta cần xác định a sao cho :
[Only registered and activated users can see links] a%7D%7D%28a%20+%201%29%20=%20%5Cfrac%7B13%7D%7B2%5 Csqrt%7Ba%20-%202%7D%7D%283%20-%20a%29
[Only registered and activated users can see links] 01%29%5E2%28a%20-%202%29%20=%20169a%283%20-%20a%29%5E2
[Only registered and activated users can see links]
[Only registered and activated users can see links] rac%7B9%7D%7B4%7D
Vậy :
[Only registered and activated users can see links] frac%7B9%7D%7B4%7Dx%5E4%20+%201%20+%20x%5E4%7D%7B2 %7D%20+%2026.%5Cfrac%7B%5Cfrac%7B1%7D%7B4%7Dx%5E4% 20+%201%20-%20x%5E4%7D%7B2%7D%20=%2016
[Only registered and activated users can see links]
Xảy ra [Only registered and activated users can see links] =%20%5Cfrac%7B2%7D%7B%5Csqrt%7B5%7D%7D
Ví dụ 4 : Tìm GTLN của hàm số :
[Only registered and activated users can see links] %206%5Csqrt%7Bx%5E3%20-%202x%5E2%20+%20x%20-%202%7D%20+%205%7D%7Bx%5E2%20+%203x%20-%204%7D
Giải :
[Only registered and activated users can see links] %206%5Csqrt%7B%28x%20-%202%29%28x%5E2%20+%201%29%7D%20+%205%7D%7Bx%5E2%2 0+%203x%20-%204%7D
Đặt [Only registered and activated users can see links] (*)
Áp dụng BDT Côsi :
[Only registered and activated users can see links] %7D%7D.2.%5Csqrt%7Ba%28x%20-%202%29%28x%5E2%20+%201%29%7D%20%5Cleq%20%5Cfrac%7 B3%7D%7B%5Csqrt%7Ba%7D%7D%5Ba%28x%20-%202%29%20+%20x%5E2%20+%201%5D
Ta cần xác định a sao cho :
[Only registered and activated users can see links] rac%7B3%7D%7B%5Csqrt%7Ba%7D%7D%20+%202%20=%20b%20% 281%29%7D%20%5C%5C%7B%5Csqrt%7Ba%7D%20=%20b%20%282 %29%7D%5C%5C%7B%5Cfrac%7B3%7D%7B%5Csqrt%7Ba%7D%7D% 281%20-%202a%29%20+%205%20=%20-4b%20%283%29%7D
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links] (Do [Only registered and activated users can see links])
Thỏa mãn (3)
Thay lại vào (2) :
[Only registered and activated users can see links]
Thay vào (*) : [Only registered and activated users can see links] 20%5Csqrt%7B5%7D%7D%7B2%7D
Vậy GTLN của hàm số là 3 .
Đạt được khi [Only registered and activated users can see links] 20%5Csqrt%7B5%7D%7D%7B2%7D .
Ví dụ 5 : (DH - B 2008)
Cho x,y là các số thực thỏa mãn : [Only registered and activated users can see links] .
Tìm GTLN và GTNN của biểu thức :
[Only registered and activated users can see links] 20+%206xy%29%7D%7B1%20+%202xy%20+%202y%5E2%7D
Lời giải :
GS k là cực trị của P ta có :
[Only registered and activated users can see links] 20+%206xy%29%7D%7B1%20+%202xy%20+%202y%5E2%7D
[Only registered and activated users can see links] %5E2%20+%206xy%29
[Only registered and activated users can see links]
Ta cần xác định k sao cho :
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
Vậy :
[Only registered and activated users can see links] ; [Only registered and activated users can see links]
Để thuần thục hơn phương pháp này các bạn làm thêm các bài tập sau :
BÀI TẬP :
1. Cho các số dương x,y thỏa mãn : [Only registered and activated users can see links]
Tìm GTNN của biểu thức :
[Only registered and activated users can see links] %204%7D%7B4x%7D%20+%20%5Cfrac%7B2%20+%20y%5E3%7D%7 By%5E2%7D
2. Cho a,b là các số dương thỏa mãn :
[Only registered and activated users can see links] %5Cfrac%7B1%7D%7B2b%7D%20=%201
Tìm GTNN của biểu thức :
[Only registered and activated users can see links]
3. Tìm GLNN của hàm số :
[Only registered and activated users can see links]
4. Tìm GTNN của hàm số :
[Only registered and activated users can see links] với [Only registered and activated users can see links]
5. Tìm GTLN của hàm số :
[Only registered and activated users can see links] rt%7B2008%20-%20x%5E2%7D%29
6. Tìm GTNN của hàm số :
[Only registered and activated users can see links] 04%5Csqrt%7B1%20-%20%5Cfrac%7Bx%7D%7B2%7D%7D
7. Cho x,y là các số không âm thỏa mãn :
[Only registered and activated users can see links] 01
Tìm GTLN của biểu thức :
[Only registered and activated users can see links] %20%5Csqrt%7By%7D
onthi.com
Ví dụ 1 : Cho x,y >= 0 thỏa mãn [Only registered and activated users can see links] 01
Tìm GTLN của biểu thức :
[Only registered and activated users can see links] 02%5Csqrt%7By%7D
Giải :
Đặt [Only registered and activated users can see links] 0a%20%5C%5C%7B%20y%20=%20b%7D
Áp dụng BDT Côsi cho 6 số :
[Only registered and activated users can see links] 3%20+%20a%5E3%20+%20a%5E3%20+%20a%5E3%20%5Cgeq%206 %5Csqrt%5B6%5D%7Bx%5E3.a%5E%7B15%7D%7D
[Only registered and activated users can see links] 3%20+%20b%5E3%20+%20b%5E3%20+%20b%5E3%20%5Cgeq%206 %5Csqrt%5B6%5D%7By%5E3.b%5E%7B15%7D%7D
Cộng vế theo vế :
[Only registered and activated users can see links] a%5E3%20+%20b%5E3%29%20%5Cgeq%206%28%5Csqrt%5B6%5D %7Ba%5E%7B15%7D%7D%5Csqrt%7Bx%7D%20+%20%5Csqrt%5B6 %5D%7Bb%5E%7B15%7D%7D.%5Csqrt%7By%7D%29%20=%206%28 %5Csqrt%7Ba%5E%7B5%7D%7D%5Csqrt%7Bx%7D%20+%20%5Csq rt%7Bb%5E%7B5%7D%7D.%5Csqrt%7By%7D%29%20%284%29
Vậy ta cần xác định a,b thỏa hệ :
[Only registered and activated users can see links] 0+%20b%5E3%20=%201%20%281%29%20%5C%5C%20%7B%5Csqrt %7Bb%5E5%7D%20=%202%5Csqrt%7Ba%5E5%7D%282%29%7D
Từ (2) :[Only registered and activated users can see links] B4%7D
thay vào (1) :
[Only registered and activated users can see links] %5D%7B64%7D%20=%201
[Only registered and activated users can see links] rac%7B1%7D%7B%5Csqrt%5B3%5D%7B1%20+%20%5Csqrt%5B5% 5D%7B64%7D%7D%7D
[Only registered and activated users can see links] 5Cfrac%7B%5Csqrt%5B5%5D%7B4%7D%7D%7B%5Csqrt%5B3%5D %7B1%20+%5Csqrt%5B5%5D%7B64%7D%20%7D%7D
Thay vào (4) :
[Only registered and activated users can see links] E3%20+%205%28a%5E3%20+%20b%5E3%29%20%5Cgeq%206%5Cs qrt%5B6%5D%7B%5Cfrac%7B1%7D%7B%281%20+%20%5Csqrt%5 B5%5D%7B64%7D%29%5E5%7D%7D%28%20%5Csqrt%7Bx%7D%20+ %202%5Csqrt%7By%7D%29
[Only registered and activated users can see links] 0%5Csqrt%5B6%5D%7B%5Cfrac%7B1%7D%7B%281%20+%20%5Cs qrt%5B5%5D%7B64%7D%29%5E5%7D%7D.P
[Only registered and activated users can see links] 0%5Csqrt%5B6%5D%7B%281%20+%20%5Csqrt%5B5%5D%7B64%7 D%29%5E5%7D
Ví dụ 2 : Tìm GTNN của hàm số :
[Only registered and activated users can see links] 5Csqrt%7B3%7D%7D%20-%20%5Csqrt%7Bx%28x%20-%201%29%7D với [Only registered and activated users can see links]
Giải :
Đặt [Only registered and activated users can see links] 20-%20a%7D
Áp dụng BDT Côsi :
[Only registered and activated users can see links] a%7D%7D.2%5Csqrt%7Bax%28x%20-%201%29%7D%20%5Cleq%20%5Cfrac%7B1%7D%7B2%5Csqrt%7B a%7D%7D.%28ax%20+%20x%20-%201%29
Ta xác định a sao cho :
[Only registered and activated users can see links] a%7D%7D%28a%20+%201%29%20=%20%5Cfrac%7B7%7D%7B4%5C sqrt%7B3%7D%7D
[Only registered and activated users can see links] 0+%201%29%5E2%20=%2049a
[Only registered and activated users can see links] 20-%2025a%20+%2012%20=%200
[Only registered and activated users can see links] %5Cfrac%7B3%7D%7B4%7D (vì [Only registered and activated users can see links])
Vậy :
[Only registered and activated users can see links] 7B%5Csqrt%7B3%7D%7D
[Only registered and activated users can see links] %7B%5Csqrt%7B3%7D%7D
Xảy ra [Only registered and activated users can see links] 4
Ví dụ 3 : Tìm GTLN của hàm số :
[Only registered and activated users can see links] +%20x%5E4%7D%20+%2013x%5E2.%5Csqrt%7B1%20-%20x%5E4%7D
Giải :
Đặt [Only registered and activated users can see links]
[Only registered and activated users can see links]
Áp dụng BDT Côsi :
[Only registered and activated users can see links] a%7D%7D2%5Csqrt%7Ba%7Dx%5E2%5Csqrt%7B1%20+%20x%5E4 %7D%20%5Cleq%20%5Cfrac%7B9%7D%7B2%5Csqrt%7Ba%7D%7D %28ax%5E4%20+%201%20+%20x%5E4%29
[Only registered and activated users can see links] Ba%20-%202%7D%7D2%5Csqrt%7Ba%20-%202%7Dx%5E2%5Csqrt%7B1%20-%20x%5E4%7D%20%5Cleq%20%5Cfrac%7B13%7D%7B2%5Csqrt% 7Ba%20-%202%7D%7D%5B%28a%20-%202%29x%5E4%20+%201%20-%20x%5E4%5D
Ta cần xác định a sao cho :
[Only registered and activated users can see links] a%7D%7D%28a%20+%201%29%20=%20%5Cfrac%7B13%7D%7B2%5 Csqrt%7Ba%20-%202%7D%7D%283%20-%20a%29
[Only registered and activated users can see links] 01%29%5E2%28a%20-%202%29%20=%20169a%283%20-%20a%29%5E2
[Only registered and activated users can see links]
[Only registered and activated users can see links] rac%7B9%7D%7B4%7D
Vậy :
[Only registered and activated users can see links] frac%7B9%7D%7B4%7Dx%5E4%20+%201%20+%20x%5E4%7D%7B2 %7D%20+%2026.%5Cfrac%7B%5Cfrac%7B1%7D%7B4%7Dx%5E4% 20+%201%20-%20x%5E4%7D%7B2%7D%20=%2016
[Only registered and activated users can see links]
Xảy ra [Only registered and activated users can see links] =%20%5Cfrac%7B2%7D%7B%5Csqrt%7B5%7D%7D
Ví dụ 4 : Tìm GTLN của hàm số :
[Only registered and activated users can see links] %206%5Csqrt%7Bx%5E3%20-%202x%5E2%20+%20x%20-%202%7D%20+%205%7D%7Bx%5E2%20+%203x%20-%204%7D
Giải :
[Only registered and activated users can see links] %206%5Csqrt%7B%28x%20-%202%29%28x%5E2%20+%201%29%7D%20+%205%7D%7Bx%5E2%2 0+%203x%20-%204%7D
Đặt [Only registered and activated users can see links] (*)
Áp dụng BDT Côsi :
[Only registered and activated users can see links] %7D%7D.2.%5Csqrt%7Ba%28x%20-%202%29%28x%5E2%20+%201%29%7D%20%5Cleq%20%5Cfrac%7 B3%7D%7B%5Csqrt%7Ba%7D%7D%5Ba%28x%20-%202%29%20+%20x%5E2%20+%201%5D
Ta cần xác định a sao cho :
[Only registered and activated users can see links] rac%7B3%7D%7B%5Csqrt%7Ba%7D%7D%20+%202%20=%20b%20% 281%29%7D%20%5C%5C%7B%5Csqrt%7Ba%7D%20=%20b%20%282 %29%7D%5C%5C%7B%5Cfrac%7B3%7D%7B%5Csqrt%7Ba%7D%7D% 281%20-%202a%29%20+%205%20=%20-4b%20%283%29%7D
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links] (Do [Only registered and activated users can see links])
Thỏa mãn (3)
Thay lại vào (2) :
[Only registered and activated users can see links]
Thay vào (*) : [Only registered and activated users can see links] 20%5Csqrt%7B5%7D%7D%7B2%7D
Vậy GTLN của hàm số là 3 .
Đạt được khi [Only registered and activated users can see links] 20%5Csqrt%7B5%7D%7D%7B2%7D .
Ví dụ 5 : (DH - B 2008)
Cho x,y là các số thực thỏa mãn : [Only registered and activated users can see links] .
Tìm GTLN và GTNN của biểu thức :
[Only registered and activated users can see links] 20+%206xy%29%7D%7B1%20+%202xy%20+%202y%5E2%7D
Lời giải :
GS k là cực trị của P ta có :
[Only registered and activated users can see links] 20+%206xy%29%7D%7B1%20+%202xy%20+%202y%5E2%7D
[Only registered and activated users can see links] %5E2%20+%206xy%29
[Only registered and activated users can see links]
Ta cần xác định k sao cho :
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
Vậy :
[Only registered and activated users can see links] ; [Only registered and activated users can see links]
Để thuần thục hơn phương pháp này các bạn làm thêm các bài tập sau :
BÀI TẬP :
1. Cho các số dương x,y thỏa mãn : [Only registered and activated users can see links]
Tìm GTNN của biểu thức :
[Only registered and activated users can see links] %204%7D%7B4x%7D%20+%20%5Cfrac%7B2%20+%20y%5E3%7D%7 By%5E2%7D
2. Cho a,b là các số dương thỏa mãn :
[Only registered and activated users can see links] %5Cfrac%7B1%7D%7B2b%7D%20=%201
Tìm GTNN của biểu thức :
[Only registered and activated users can see links]
3. Tìm GLNN của hàm số :
[Only registered and activated users can see links]
4. Tìm GTNN của hàm số :
[Only registered and activated users can see links] với [Only registered and activated users can see links]
5. Tìm GTLN của hàm số :
[Only registered and activated users can see links] rt%7B2008%20-%20x%5E2%7D%29
6. Tìm GTNN của hàm số :
[Only registered and activated users can see links] 04%5Csqrt%7B1%20-%20%5Cfrac%7Bx%7D%7B2%7D%7D
7. Cho x,y là các số không âm thỏa mãn :
[Only registered and activated users can see links] 01
Tìm GTLN của biểu thức :
[Only registered and activated users can see links] %20%5Csqrt%7By%7D
onthi.com