\(\left(\frac{1}{3}-2x\right)^{2018}+\left(3y-x\right)^{2020}\le0\)(1)

Vì \(\left(\frac{1}{3}-2x\right)^{2018}\ge0\forall x\); \(\left(3y-x\right)^{2020}\ge0\forall x,y\)

\(\Rightarrow\left(\frac{1}{3}-2x\right)^{2018}+\left(3y-x\right)^{2020}\ge0\forall x,y\)(2)

Từ (1), (2) \(\Rightarrow\left(\frac{1}{3}-2x\right)^{2018}+\left(3y-x\right)^{2020}=0\)

\(\Leftrightarrow\hept{\begin{cases}\frac{1}{3}-2x=0\\3y-x=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=\frac{1}{6}\\y=\frac{1}{18}\end{cases}}\)

\(\Rightarrow\frac{1}{x}+\frac{1}{y}=6+18=24\left(đpcm\right)\)

Ta có: \(\left(\dfrac{1}{3}-2x\right)^{2018}\ge0\forall x\);

\(\left(3y-x\right)^{2020}\ge0\forall x;y\)

=> \(\left(\dfrac{1}{3}-2x\right)^{2018}+\left(3y-x\right)^{2020}\ge0\)

mà theo đề thì:\(\left(\dfrac{1}{3}-2x\right)^{2018}+\left(3y-x\right)^{2020}\le0\)

=> Dấu ''='' xảy ra khi và chỉ khi: \(\left\{{}\begin{matrix}\dfrac{1}{3}-2x=0\\3y-x=0\end{matrix}\right.\)

Ta có: \(\dfrac{1}{3}-2x=0\Rightarrow x=\dfrac{1}{6}\);

\(3y-x=0\Leftrightarrow3y-\dfrac{1}{6}=0\Leftrightarrow3y=\dfrac{1}{6}\Leftrightarrow y=\dfrac{1}{18}\)

=> \(\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{\dfrac{1}{6}}+\dfrac{1}{\dfrac{1}{18}}=6+18=24\left(đpcm\right)\)

thanks bn

đúng thì mk tick

 Ko biết Anh gì ơi

Nhận xét : ( x + y - 3 )^2018 >=0 và 2018.(2x-4)^2020 >= 0

=> (x+y-3)^2018 + 2018.(2x-4)^2020 >=0 

Dấu = xảy ra khi : x + y - 3 = 0 và 2x - 4 = 0 => x = 2 và y = 1

Thay vào bt S :

S = ( 2 - 1)^2019 + (2-1)^2019

= 1^2019 + 1^2019 = 2

em cảm ơn

\(5x^2+5y^2+8xy-2x+2y+2=0\)

\(\Leftrightarrow4\left(x+y\right)^2+\left(x-1\right)^2+\left(y+1\right)^2=0\)

Vì \(\left(x+y\right)^2\ge0,\left(x-1\right)^2\ge0,\left(y+1\right)^2\ge0\)

\(\Rightarrow4\left(x+y\right)^2+\left(x-1\right)^2+\left(y+1\right)^2\ge0\)

Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}x+y=0\\x-1=0\\y+1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=-1\end{matrix}\right.\)

\(\left(x+y\right)^{2018}+\left(x-2\right)^{2019}+\left(y+1\right)^{2020}=\left(1-1\right)^{2018}+\left(1-2\right)^{2019}+\left(-1+1\right)^{2020}=-1\)

\(\left(2x-5\right)^{2018}+\left(3y+4\right)^{2020}\le0\\ \Leftrightarrow\left(2x-5\right)^{2018}+\left(3y+4\right)^{2020}=0\\ \Leftrightarrow\left\{{}\begin{matrix}\left(2x-5\right)^{2018}=0\\\left(3y+4\right)^{2020}=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{5}{2}\\y=-\dfrac{4}{3}\end{matrix}\right.\\ \Leftrightarrow M=6x^2+9xy-y^2-5x^2+2xy=x^2+11xy-y^2\\ \Leftrightarrow M=\dfrac{25}{4}-11\cdot\dfrac{4}{3}\cdot\dfrac{5}{2}-\dfrac{16}{9}=\dfrac{25}{4}-\dfrac{110}{3}-\dfrac{16}{9}=-\dfrac{1159}{36}\)

Em cảm ơn.

\(\dfrac{1}{x}+\dfrac{2}{y}+\dfrac{3}{z}=0\)

=>\(\dfrac{yz+2xz+3xy}{xyz}=0\)

=>yz+2xz+3xy=0

=>\(xy+\dfrac{2}{3}xz+\dfrac{1}{3}yz=0\)

\(x+\dfrac{y}{2}+\dfrac{z}{3}=1\)

=>\(\left(x+\dfrac{y}{2}+\dfrac{z}{3}\right)^2=1\)

=>\(x^2+\dfrac{y^2}{4}+\dfrac{z^2}{9}+2\left(x\cdot\dfrac{y}{2}+x\cdot\dfrac{z}{3}+\dfrac{y}{2}\cdot\dfrac{z}{3}\right)=1\)

=>\(A+2\left(\dfrac{xy}{2}+\dfrac{xz}{3}+\dfrac{yz}{6}\right)=1\)

=>A+xy+2/3xz+1/3yz=1

=>A=1