Ta có :

aaa : 37 .x y = a

a x 111 : 37 x y = a

a x 3 x y = a

=> 3 x y = 1

=> y = 1/3

Ta có:

aaa : 37 x y =a 

a x 111 : 37 x y = a

111 : 37 x y = a : a

3 x y =1 

y=1/3

Ta có:

aaa : 37 x y =a 

a x 111 : 37 x y = a

111 : 37 x y = a : a

3 x y =1 

y=1/3

\(x^2+y^2=x+y\\ \Leftrightarrow x^2-x+y^2-y=0\\ \Leftrightarrow\left(x-\dfrac{1}{2}\right)^2+\left(y-\dfrac{1}{2}\right)^2=\dfrac{1}{2}\\ A=x+y=\left(x-\dfrac{1}{2}\right)+\left(y-\dfrac{1}{2}\right)+1\)

Áp dụng Bunhiacopski:

\(\left[\left(x-\dfrac{1}{2}\right)+\left(y-\dfrac{1}{2}\right)\right]^2\le\left(1^2+1^2\right)\left[\left(x-\dfrac{1}{2}\right)^2+\left(y-\dfrac{1}{2}\right)^2\right]=2\cdot\dfrac{1}{2}=1\\ \Leftrightarrow A\le1+1=2\)\(A_{max}=2\Leftrightarrow x=y=1\)

\(x^2+y^2\ge0\Rightarrow x+y=x^2+y^2\ge0\)

\(A_{min}=0\) khi \(x=y=0\)

a) \(6xy+4x-9y-7=0\)

  \(\Leftrightarrow2x.\left(3y+2\right)-9y-6-1=0\)

\(\Leftrightarrow2x.\left(3y+x\right)-3.\left(3y+2\right)=1\)

\(\Leftrightarrow\left(2x-3\right).\left(3y+2\right)=1\)

Mà \(x,y\in Z\Rightarrow2x-3;3y+2\in Z\)

Tự làm típ

\(A=x^3+y^3+xy\)

\(A=\left(x+y\right)\left(x^2-xy+y^2\right)+xy\)

\(A=x^2-xy+y^2+xy\)( vì \(x+y=1\))

\(A=x^2+y^2\)

Áp dụng bất đẳng thức Bunhiakovxky ta có :

\(\left(1^2+1^2\right)\left(x^2+y^2\right)\ge\left(x\cdot1+y\cdot1\right)^2=\left(x+y\right)^2=1\)

\(\Leftrightarrow2\left(x^2+y^2\right)\ge1\)

\(\Leftrightarrow x^2+y^2\ge\frac{1}{2}\)

Hay \(x^3+y^3+xy\ge\frac{1}{2}\)

Dấu "=" xảy ra \(\Leftrightarrow x=y=\frac{1}{2}\)

Đề là: \(Q=\left(1+\dfrac{a}{x}\right)\left(1+\dfrac{a}{y}\right)\left(1+\dfrac{a}{z}\right)\) đúng không em nhỉ?

Ta có:

\(Q=\left(1+\dfrac{x+y+z}{x}\right)\left(1+\dfrac{x+y+z}{y}\right)\left(1+\dfrac{x+y+z}{z}\right)\)

\(=\dfrac{\left(x+x+y+z\right)\left(x+y+y+z\right)\left(x+y+z+z\right)}{xyz}\)

\(Q\ge\dfrac{4\sqrt[4]{x^2yz}.4\sqrt[4]{xy^2z}.4\sqrt[4]{xyz^2}}{xyz}=\dfrac{64xyz}{xyz}=64\)

\(Q_{min}=64\) khi \(x=y=z=\dfrac{a}{3}\)

\(A=\dfrac{1}{x}+\dfrac{1}{4y}=\dfrac{4}{4x}+\dfrac{1}{4y}=\dfrac{2^2}{4x}+\dfrac{1^2}{4y}\)

Áp dụng BĐT Cauchy schwart, ta có:

\(A=\dfrac{2^2}{4x}+\dfrac{1^2}{4y}\ge\dfrac{\left(2+1\right)^2}{4\left(x+y\right)}=\dfrac{9}{4.2}=\dfrac{9}{8}\)

Dấu "=" xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{2}{4x}=\dfrac{1}{4y}\\x+y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{2x}=\dfrac{1}{4y}\\x+y=2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2x=4y\\x+y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=2y\\x+y=2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{4}{3}\\y=\dfrac{2}{3}\end{matrix}\right.\)

Vậy, GTNN của \(A=\dfrac{9}{8}\Leftrightarrow\left(x,y\right)=\left(\dfrac{4}{3},\dfrac{2}{3}\right)\)

Áp dụng BĐT Cosi cho 2 cặp số dương là  \(\dfrac{1}{x};\dfrac{9}{16}x\) và \(\dfrac{1}{4y};\dfrac{9}{16}y\) , ta có:

\(\dfrac{1}{x}+\dfrac{9}{16}x\ge2\sqrt{\dfrac{1}{x}.\dfrac{9}{16}x}=2.\dfrac{3}{4}=\dfrac{3}{2}\)

\(\dfrac{1}{4y}+\dfrac{9}{16}y\ge2\sqrt{\dfrac{1}{4y}.\dfrac{9}{16}y}=2.\dfrac{3}{8}=\dfrac{3}{4}\)

Cộng vế theo vế ta được: \(\dfrac{1}{x}+\dfrac{1}{4y}+\dfrac{9}{16}\left(x+y\right)\ge\dfrac{3}{2}+\dfrac{3}{4}=\dfrac{9}{4}\)

\(\Leftrightarrow A+\dfrac{9}{16}.2\ge\dfrac{9}{4}\Leftrightarrow A\ge\dfrac{9}{4}-\dfrac{9}{8}=\dfrac{9}{8}\)

Dấu bằng xảy ra \(\Leftrightarrow\left(x,y\right)=\left(\dfrac{4}{3};\dfrac{2}{3}\right)\)