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\(\left(2x-5\right)^{2018}+\left(3y+4\right)^{2020}\le0\\ \Leftrightarrow\left(2x-5\right)^{2018}+\left(3y+4\right)^{2020}=0\\ \Leftrightarrow\left\{{}\begin{matrix}\left(2x-5\right)^{2018}=0\\\left(3y+4\right)^{2020}=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{5}{2}\\y=-\dfrac{4}{3}\end{matrix}\right.\\ \Leftrightarrow M=6x^2+9xy-y^2-5x^2+2xy=x^2+11xy-y^2\\ \Leftrightarrow M=\dfrac{25}{4}-11\cdot\dfrac{4}{3}\cdot\dfrac{5}{2}-\dfrac{16}{9}=\dfrac{25}{4}-\dfrac{110}{3}-\dfrac{16}{9}=-\dfrac{1159}{36}\)
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M=6x^2+9xy-y^2-5x^2+2xy=x^2+11xy-y^2
(2x-5)^2020+(3y+4)^2022<=0
=>x=5/2 và y=-4/3
M=25/4+11*5/2*(-4/3)-16/9=-1159/36
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\(\left(x+1\right)^{2020}+\left(2-3y\right)^{2022}=0\)
Vì \(\hept{\begin{cases}\left(x+1\right)^{2020}\ge0\forall x\\\left(2-3y\right)^{2022}\ge0\forall y\end{cases}}\Rightarrow\left(x+1\right)^{2020}+\left(2-3y\right)^{2022}\ge0\forall x,y\)
Dấu " = " xảy ra khi và chỉ khi \(\hept{\begin{cases}\left(x+1\right)^{2020}=0\\\left(2-3y\right)^{2022}=0\end{cases}}\Rightarrow\hept{\begin{cases}x=-1\\3y=2\end{cases}}\Rightarrow\hept{\begin{cases}x=-1\\y=\frac{2}{3}\end{cases}}\)
( x + 1 )2020 + ( 2 - 3y )2022 = 0
Ta có \(\hept{\begin{cases}\left(x+1\right)^{2020}\ge0\forall x\\\left(2-3y\right)^{2022}\ge0\forall y\end{cases}}\Rightarrow\left(x+1\right)^{2020}+\left(2-3y\right)^{2022}\ge0\forall x,y\)
Đẳng thức xảy ra <=> \(\hept{\begin{cases}x+1=0\\2-3y=0\end{cases}}\Rightarrow\hept{\begin{cases}x=-1\\y=\frac{2}{3}\end{cases}}\)
Vậy x = -1 ; y = 2/3
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....Bạn viết rõ đề ra được không ? Mình nhìn đề không hiểu lắm
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Sửa: \(\left(\dfrac{1}{3}-2x\right)^{2020}+\left(3y-x\right)^{2022}\le0\)
Mà \(\left(\dfrac{1}{3}-2x\right)^{2020}+\left(3y-x\right)^{2022}\ge0\) với mọi x,y
Do đó \(\left\{{}\begin{matrix}\dfrac{1}{3}-2x=0\\3y-x=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{1}{6}\\y=\dfrac{1}{18}\end{matrix}\right.\)
\(\Leftrightarrow\dfrac{1}{x}+\dfrac{1}{y}=6+18=24\)