a) \(M=\frac{x^4+2}{x^6+1}+\frac{x^2-1}{x^4-x^2+1}+\frac{x^2+3}{x^4+4x^2+3}\)

\(M=\frac{x^4+2}{\left(x^2+1\right)\left(x^4-x^2+1\right)}+\frac{x^2-1}{x^4-x^2+1}-\frac{x^2+3}{x^4+3x^2+x^2+3}\)

\(M=\frac{x^4+2}{\left(x^2+1\right)\left(x^4-x^2+1\right)}+\frac{x^2-1}{x^4-x^2+1}-\frac{x^2+3}{x^2\left(x^2+3\right)+x^2+3}\)

\(M=\frac{x^4+2}{\left(x^2+1\right)\left(x^4-x^2+1\right)}+\frac{x^2-1}{x^4-x^2+1}-\frac{x^2+3}{\left(x^2+3\right)\left(x^2+1\right)}\)

\(M=\frac{x^4+2}{\left(x^2+1\right)\left(x^4-x^2+1\right)}+\frac{x^2-1}{x^4-x^2+1}-\frac{1}{x^2+1}\)

\(M=\frac{x^4+2+x^4-1-x^4+x^2-1}{\left(x^2+1\right)\left(x^4-x^2+1\right)}\)

\(M=\frac{0+x^4+x^2}{\left(x^2+1\right)\left(x^4-x^2+1\right)}\)

\(M=\frac{x^2\left(x^2+1\right)}{\left(x^2+1\right)\left(x^4-x^2+1\right)}\)

\(M=\frac{x^2}{x^4-x^2+1}\)

Đặt : \(a=2^x;b=2^y;c=2^z\)

Khi đó :  \(a,b,c>0;abc=2^{x+y+z}=64\)

Ta cần c/m : \(a^3+b^3+c^3\ge4\left(a^2+b^2+c^2\right)\)

\(\Rightarrow a^3+32-6a^2=\left(a-4\right)^2\left(a+2\right)\)

Theo đó, ta cần sử dụng giả thiết : \(a>0\), suy ra : \(a^3+32\ge6a^2\)

Thiết lập các bđt tương tự cho b và c và cộng theo vế các bđt tìm được, ta có :

\(a^3+b^3+c^3+96\ge6\left(a^2+b^2+c^2\right)\)

Ta cần c/m thêm : \(6\left(a^2+b^2+c^2\right)\ge4\left(a^2+b^2+c^2\right)+96\)

hay : \(2\left(a^2+b^2+c^2\right)\ge2.3\sqrt[3]{a^2b^2c^2}=6\sqrt[3]{4096}=96\)

\(\Rightarrowđpcm\)

mik làm cách khác,mấy bạn cho điểm nhá!

Sai đề:x+y+z=6

Đặt\(a=2^x,b=2^y,c=2^z\)

\(\Rightarrow abc=2^{x+y+z}=64\)

Áp dụng bất đẳng thức AM-GM,ta được:

\(3\sqrt[3]{abc}\le a+b+c\)

Ta có:\(3\left(a^3+b^3+c^3\right)\ge\left(a+b+c\right)\left(a^2+b^2+c^2\right)\)

Hay \(2\left(a^3+b^3+c^3\right)\ge ab\left(a+b\right)+bc\left(b+c\right)+ca\left(c+a\right)\)

Thật vậy:

Áp dụng bất đẳng thức AM-GM một lần nữa,ta được:

\(a^3+a^3+b^3\ge3a^2b\)

\(a^3+a^3+c^3\ge3a^2c\)

\(a^3+b^3+b^3\ge3b^2a\)

\(a^3+c^3+c^3\ge3c^2a\)

\(b^3+b^3+c^3\ge3b^2c\)

\(b^3+c^3+c^3\ge3c^2b\)

Cộng vế theo vế của các bất đẳng thức,ta được:

\(2\left(a^3+b^3+c^3\right)\ge ab\left(a+b\right)+bc\left(b+c\right)+ca\left(c+a\right)\)

Dấu "="xẩy ra khi và chỉ khi:\(a=b=c\)

Đặt biểu thức là A

\(2A=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+.2018.2019\)

\(2A=\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+...+\left(\frac{1}{2017.2018}-\frac{1}{2018.2019}\right)\)

\(2A=\frac{1}{2}-\frac{1}{2018.2019}\)

A= 1/4 - 1/(2018.2019)

Vậy A = ... (tự ghi)

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{2017.2018.2019}\)

\(=\frac{1}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{2017.2018.2019}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{2017.2018}-\frac{1}{2018.2019}\right)\)

\(=\frac{1}{2}.\left(1-\frac{1}{2018.2019}\right)\)

Tự làm nốt

\(\left(x-2012\right)^2+\left(x+2013\right)^2\)

\(=x^2-2.2012+2012^2+x^2+2.2013+2013^2\)

\(=2x^2+2x+2012^2+2013^2\)

\(=2\left(x^2+x+\frac{1}{4}\right)+8100312,5\)

\(=2\left(x+\frac{1}{2}\right)^2+8100312,5\)

Bí 

alibaba

kudo shinichi sai oy

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11, If I had eaten lunch , I would feel hungry now

12, But for rain , we would have a better crop

13,  Unless they invite me, I won’t come

13,  If he had revised all his lessons, he wouldn’t have failed the exam

14, But for his sister’s money, he wouldn’t have continued to study

15, If you walk to the park, it will take you 5 minutes

16, If it rains, we’ll stay at home

17, If I were you, I would cut down on smoking right now 

18, Unless you lave me alone , I'll call the police 

19, Should you arrive at the ofice earlier than I do, please turn on the air-conditioner

20, If it snows , the children won't go to school 

21, Had he not died so young , he would have been a famous musician by now 

22, Unless you tell me the whole truth , I won't help you

23, Were I you , I would take good care of it

bài có 2 câu 13 nhé bạn !

11 i didn't eat lunch , i feel hungry now

=>if i had eaten lunch, I wouldn't have been hungry now

12 it hadn't rained , we would have a better crop

=>but for the rain, we would have a better crop

13 i only come if they invite me

=> unless they invite, I won't come

13 he didn't revise all his lessons , he failed the exam

=>if he had rivised all his lessons, he wouldn't have failed the exam

14 his sister sent her money , so he continued to study

=>but for his sister's money, he wouldn't have contrinued to study

15 the park is over there , only 5 minutes walk

=>if you walk for 5 minutes, you will find the park

16 in case it rain , we'll stay at home

=> if it rains, we'll stay at home

17 let me give you my advice . you should cut down on smoking right now

=> if i were you, I would cut down on smoking right now

18leave me alone or i'll call the police

=>unless you leave me alone, I won't call the police

19 if you arrive at the office earlier than i do , please turn on the air -conditioner

=> should you arrive at the office earlier than I do, please turn on the air-conditioner

\20 the children don't go to school in the snowy weather

=>if it doesn't snow, the children will go to school

21` he died so young otherwise , he would be a famous musician by now

=> had it not been for his dead, he would have been a famous musician by now

22 you must tell me the whole truth or i won't help you

=> unless you tell me the whole truth, I will help you

23 the car breaks down so often because you don't take good care of it

=> were i take good care of it, the car wouldn't break down so often

Nếu không hiểu câu nào thì bình luận ngay ở đây để mình giải thích chi tiết nhé.

Thằng dog kia mày cho nhiều câu hỏi thế sao tao trả lời -_-*

1 he can not go out because he has to study for his exam

=> if he doesn't have to study for the exam, he can go out.

2 she is lazy so she can not pass the exam

=> if she isn't lazy, she can pass the exam.

3 he will pay me tonight , i will have enough money to buy a car

=> if he pays me tonight, I will have enough money to buy a car.

4 he smokes too much , that is why he can not get rid of his cough

=>if he doesn't smoke too much, he can not get rid of his cough.

5 she is very shy , so she does not enjoy the pary

=>if she isn't shy, she will enjoy the party.

(Tớ làm 5 câu thôi, mấy câu sau cậu tự làm nhé)