1.cho a, b, c dương. CMR

\(\sqrt{a^2-ab+b^2}\) + \(\sqrt{b^2-bc+c^2}\) \(\ge\) \(\sqrt{a^2+ac+c^2}\)

2.Cho a, b, c, d dương. CMR

\(\sqrt{a^2-\sqrt{2}ab+b^2}\) + \(\sqrt{b^2-\sqrt{2}bc+c^2}\) \(\ge\) \(\sqrt{a^2+c^2}\)

3.Cho a, b, c, d dương. CMR

\(\sqrt{a^2-\sqrt{3}ab+b^2}\) + \(\sqrt{b^2-bc+c^2}\)+\(\sqrt{c^2-\sqrt{3}cd+d^2}\) \(\ge\) \(\sqrt{a^2+ad+d^2}\)

(DỰA THEO ĐỊNH LÝ HÀM SỐ COSIN)

Bài 1:

1. \(\sqrt{x-5}\)+ \(\sqrt{x}\)=\(\sqrt{5}\)

2.\(\frac{x^2}{\sqrt{3x-2}}\)- \(\sqrt{3x-2}\)=1-x

3.\(\sqrt{x+2\sqrt{x-1}}\) + \(\sqrt{x-2\sqrt{x-1}}\)=\(\frac{x+3}{2}\)

4. \(\sqrt{x-1}\)+\(\sqrt{3x-2}\)= 4x-9+2\(\sqrt{3x^2-5x+2}\)

5. \(\sqrt{x+\sqrt{2x-1}}\)+ \(\sqrt{x-\sqrt{2x-1}}\)=\(\sqrt{2}\)

6. x-2\(\sqrt{x-1}\) - (x - 1)\(\sqrt{x}\) + \(\sqrt{x^2-x}\)=0

7. \(\sqrt{x\left(x-1\right)}\) +\(\sqrt{x\left(x+2\right)}\)= 2\(\sqrt{x^2}\)

Câu 2 : Xét dấu các biểu thức sau :

A = \(\frac{4-3x}{2x+1}\)

B = \(1-\frac{2-x}{3x-2}\)

C = \(x\left(x-2\right)^2\left(3-x\right)\)

D = \(\frac{x\left(x-3\right)^2}{\left(x-5\right)\left(1-x\right)}\)

E = \(-x^2+x+6\)

F = \(2x^2-\left(2+\sqrt{3}\right)x+\sqrt{3}\)

G = \(\left(3x-1\right)\left(x+2\right)\)

H = \(\frac{2-3x}{5x-1}\)

K = \(\left(-x+1\right)\left(x+2\right)\left(3x+1\right)\)

L = \(2-\frac{2+x}{3x-2}\)

M = \(9x^2-1\)

N = \(-x^3+7x-6\)

O = \(x^3+x^2-5x+3\)

P = \(x^2-x-2\sqrt{2}\)

Q = \(\frac{1}{3-x}-\frac{1}{3+x}\)

R = \(\frac{x^2-6x+8}{x^2+8x-9}\)

S= \(\frac{x^2+4x+4}{x^4-2x^2}\)

T = \(\frac{\left|x+1\right|-1}{x^2+x+1}\)

Khảo sát sự biến thiên và vẽ đồ thị các hàm số sau

1 , y = 2x² - x - 2

2 , y = \(-\frac{1}{2}x^2+2x-1\)

3 , y = \(2x^2-2x\)

4 , y = \(-\frac{1}{2}x^2-x+\frac{3}{2}\)

5 , y = \(-x^2-4x+3\)

6 , y = \(-2x^2-x+2\)

7, y = \(-2x^2-2\)

8 , y = \(\left(\frac{1}{2}x^2-2x-6\right)\)

9 , y = \(2\left(x+3\right)^2\)

f(-2)=\(\frac{2}{x-1}\)

Thay x=-2 ta được : 

\(\frac{2}{-2-1}\)=\(\frac{-2}{3}\)

f(0)=\(\sqrt{x+1}\)

Thay x=0 ta được \(\sqrt{0+1}\)=1

f(2)=\(\sqrt{x+1}\)

Thay x=2 ta được \(\sqrt{2+1}\)=\(\sqrt{3}\)

f(3)=\(\frac{x^2-1}{x+1}\)

Thay x= 3 ta được 

\(\frac{3^2-1}{3+1}\)=2

4. Dấu của tam thức bậc hai

Bài 5: Giải các bất phương trình sau:

a.2x\(^2\) - 5x + 2 < 0 b. -5x\(^2\) + 4x + 12 <0 c. 16x\(^2\) + 40x + 25 >0

d. -2x\(^2\) + 3x - 7\(\ge\)0 e. 3x\(^2\) - 4x + 4 \(\ge\) 0 f. x\(^2\) - x - 6\(\le\) 0

g. \(\frac{-3x^2-x+4}{x^2+3x+5}\) > 0 h. \(\frac{x-1}{x^2-4}\) - \(\frac{2}{x+2}\) > \(\frac{1}{x-2}\) i. \(\frac{8}{x^2-9}+\frac{3}{x+3}< \frac{2}{x-3}\)

j. \(2x^2-\left|5x-3\right|< 0\) k. \(x-8>\left|x^2+3x-4\right|\) l. \(\left|x^2-1\right|-2x< 0\)

m. \(\sqrt{4-x}>2+x\) n. \(\sqrt{9-x}-3>x\)

\(\frac{a^3}{b^2}\)+\(\frac{b^3}{c^2}\)+\(\frac{c^3}{a^2}\) ≥ \(\frac{a^2}{b}\)+\(\frac{b^2}{c}\)+\(\frac{c^2}{a}\)(1)

(2) \(\frac{a^5}{b^3}\)+\(\frac{b^5}{c^3}\)+\(\frac{c^5}{a^3}\) ≥ \(a^2\)+\(b^2\)+\(c^2\)
chứng minh bằng cosi
giúp mk với :<<<<

bài 2: giải các bpt sau:

1) (x-2)(\(9-x^2\))≤0

2) (\(x^2-x-6\))(\(x^2-3x+2\))≥0

3) \(\frac{\left(x-2\right)\left(9-x\right)}{x-1}\)≤0

4) \(\frac{x\left(x^2-3x+2\right)}{x+4}\)≥0

5) \(\frac{\left(x+2\right)}{\left(x+1\right)\left(x-2\right)}\)<0

6) \(\frac{\left(x-2\right)\left(9-x^2\right)}{x-1}\)≥0

7) \(\frac{x^2\left(x-3\right)}{3x^2+x-4}\)≥0

8) \(\frac{x^2-3x+2}{9-x}\)≥0

9) \(\frac{x^2+1}{x^2+3x-10}\)≤0

10) \(\frac{x^2-9x+14}{x^2+9x+14}\)≥0

1. \(x^3-x^2+12x\sqrt{x-1}+20=0\)

2. \(x^3+\sqrt{\left(x-1\right)^3}=9x+8\)

3. \(\sqrt{2x^2+x+1}+\sqrt{x^2-x+1}=3x\)

4. \(x^6+\left(x^3-3\right)^3=3x^5-9x^2-1\)

5. \(x^2-6\left(x+3\right)\sqrt{x+1}+14x+3\sqrt{x+1}+13=0\)

6. \(x^2-4x+\left(x-3\right)\sqrt{x^2-x+1}=-1\)

7. \(\sqrt{2x-1}+\sqrt{5-x}=x-2+2\sqrt{-2x^2+11x-5}\)

8. \(\sqrt{5x+11}-\sqrt{6-x}+5x^2-14x-60=0\)

9. \(x^2+6x+8=3\sqrt{x+2}\)

10. \(2x^2+3x-2=\left(2x-1\right)\sqrt{2x^2+x-3}\)

11. \(\sqrt{x+1}+\sqrt{4-x}-\sqrt{\left(x+1\right)\left(4-x\right)}=1\)

12. \(x^2-\sqrt{x^2-4x}=4\left(x+3\right)\)

13. \(x^2-x-4=2\sqrt{x-1}\left(1-x\right)\)

14. \(\frac{1}{\sqrt{x}+1}+\frac{1}{\sqrt{x}-1}=1\)

15. \(\sqrt{2x^2+3x+2}+\sqrt{4x^2+6x+21}=11\)

16. \(\sqrt{x+3+3\sqrt{2x-3}}+\sqrt{x-1+\sqrt{2x-1}}=2\sqrt{2}\)

17. \(\left(x-2\right)^2\left(x-1\right)\left(x-3\right)=12\)

18. \(2x^2+\sqrt{x^2-2x-19}=4x+74\)

19. \(x^4+x^2-20=0\)

20. \(x+\sqrt{4-x^2}=2+3x\sqrt{4-x^2}\)

21. \(\left(x^2+x+1\right)\left(\sqrt[3]{\left(3x-2\right)^2}+\sqrt[3]{3x-2}+1\right)=9\)

22. \(\sqrt{x^2-3x+5}+x^2=3x+7\)

23. \(x^2+6x+5=\sqrt{x+7}\)

24. \(\frac{2x^2-3x+10}{x+2}=3\sqrt{\frac{x^2-2x+4}{x+2}}\)

25. \(5\sqrt{x-1}-\sqrt{x+7}=3x-4\)

26. \(2\left(x^2+2\right)=5\sqrt{x^3+1}\)

27. \(\sqrt{x-1}+\sqrt{5-x}-2=2\sqrt{\left(x-1\right)\left(5-x\right)}\)

28. \(x^2+\frac{9x^2}{\left(x-3\right)^2}=40\)

29. \(\frac{26x+5}{\sqrt{x^2+30}}+2\sqrt{26x+5}=3\sqrt{x^2+30}\)

30. \(\frac{\sqrt{27+x^2+x}}{2+\sqrt{5-\left(x^2+x\right)}}=\frac{\sqrt{27+2x}}{2+\sqrt{5-2x}}\)