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\(M=5x^2+y^2-2x+2y+2xy+2004\)
\(=\left(x^2+2x+1\right)+2y\left(x+1\right)+y^2+4x^2-4x+1+2002\)
\(=\left(x+1\right)^2+2y\left(x+1\right)+y^2+\left(2x-1\right)^2+2002\)
\(=\left(x+1+y\right)^2+\left(2x-1\right)^2+2003\ge2002\) với mọi x,y
=> \(M_{min}=2002\Leftrightarrow\left\{{}\begin{matrix}x+y+1=0\\2x-1=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}y=-\dfrac{3}{2}\\x=\dfrac{1}{2}\end{matrix}\right.\)
Vậy \(M_{min}=2002\)
\(S=\left(x^2+y^2+1+2xy+2x+2y\right)+\left(y^2-4y+4\right)+2021\)
\(S=\left(x+y+1\right)^2+\left(y-2\right)^2+2021\ge2021\)
Dấu "=" xảy ra khi \(\left(x;y\right)=\left(-3;2\right)\)
\(P=\sqrt{\frac{1}{36}\left(11a+7b\right)^2+\frac{59\left(a-b\right)^2}{36}}+\sqrt{\frac{1}{36}\left(7a+11b\right)+\frac{59\left(a-b\right)^2}{36}}\)
\(=\sqrt{\frac{1}{16}\left(3a+5b\right)^2+\frac{5\left(a-b\right)^2}{16}}+\sqrt{\frac{1}{16}\left(5a+3b\right)^2+\frac{5\left(a-b\right)^2}{16}}\)
\(\ge\frac{1}{6}\left(11a+7b\right)+\frac{1}{6}\left(7a+11b\right)+\frac{1}{4}\left(3a+5b\right)+\frac{1}{4}\left(5a+3b\right)\)
\(=5\left(a+b\right)=5.2016=10080\)
Ta có:
\(2x^2+xy+2y^2=x^2+y^2+\frac{3}{4}\left(x+y\right)^2+\frac{1}{4}\left(x-y\right)^2\)
\(\ge\frac{2\left(x+y\right)^2}{4}+\frac{3\left(x+y\right)^2}{4}=\frac{5\left(x+y\right)^2}{4}\)
\(\Rightarrow\sqrt{2x^2+xy+2y^2}\ge\frac{\sqrt{5}}{2}\left(x+y\right)\). Tương tự ta có:
\(\sqrt{2y^2+yz+2z^2}\ge\frac{\sqrt{5}}{2}\left(y+z\right);\sqrt{2z^2+xz+2x^2}\ge\frac{\sqrt{5}}{2}\left(x+z\right)\)
\(\Rightarrow M\ge\frac{\sqrt{5}}{2}\left(x+y\right)+\frac{\sqrt{5}}{2}\left(y+z\right)+\frac{\sqrt{5}}{2}\left(x+z\right)\)
\(=\sqrt{5}\left(x+y+z\right)=\sqrt{5}\)
Đẳng thức xảy ra khi \(x=y=z=\frac{1}{3}\)
Cho mình hối tại sao đẳng thức sảy ra x=y=z=1/3 vậy
Ta có A = 5x2 - 2xy + 2y2 - 4x + 2y + 3
=> 2A = 10x2 - 4xy + 4y2 - 8x + 4y + 6
= (x2 - 4xy + 4y2) - 2(x - 2y) + 1 + 9x2 - 6x + 1 + 4
= \(\left(x-2y\right)^2-2\left(x-2y\right)+1+9\left(x^2-\frac{2}{3}x+\frac{1}{9}\right)+4\)
\(=\left(x-2y-1\right)^2+9\left(x-\frac{1}{3}\right)^2+4\)\(\ge4\)
=> A \(\ge\)2
Dấu "=" xảy ra <=> \(\hept{\begin{cases}x-2y-=0\\x-\frac{1}{3}=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x-2y=1\\x=\frac{1}{3}\end{cases}}\Leftrightarrow\hept{\begin{cases}y=-\frac{1}{3}\\x=\frac{1}{3}\end{cases}}\)
Vậy khi x = 1/3 ; y = -1/3 thì A đạt GTNN
\(A=5x^2+2y^2-2xy-4x+2y\)\(+3\)
\(=\left(x^2-2xy+y^2\right)+\)\(\left(4x^2-4x+1\right)+\)\(\left(y^2+2y+1\right)+1\)
\(Tacó\)
\(y\ge xy+1\ge2\sqrt{xy}\Rightarrow\sqrt{\dfrac{y}{x}}\ge2\Rightarrow\dfrac{y}{x}\ge4\)
\(Q=\dfrac{1-\dfrac{2y}{x}+2\left(\dfrac{y}{x}\right)^2}{\dfrac{y}{x}+\left(\dfrac{y}{x}\right)^2}\)
Đặt \(\dfrac{y}{x}=a\ge4\)
\(Q=\dfrac{2a^2-2a+1}{a^2+a}=\dfrac{2a^2-2a+1}{a^2+a}-\dfrac{5}{4}+\dfrac{5}{4}=\dfrac{\left(a-4\right)\left(3a-1\right)}{4\left(a^2+1\right)}+\dfrac{5}{4}\ge\dfrac{5}{4}\)
\(Q_{min}=\dfrac{5}{4}\) khi \(a=4\) hay \(\left(x;y\right)=\left(\dfrac{1}{2};2\right)\)
\(M=y^2+2y\left(x+1\right)+\left(x+1\right)^2-\left(x+1\right)^2+5x^2-2x+2016\)
\(M=\left(y+x+1\right)^2+4x^2-4x+1+2014\)
\(M=\left(y+x+1\right)^2+\left(2x-1\right)^2+2014\)
Dễ thấy \(\left(y+x+1\right)^2\ge0\forall x;y\)và \(\left(2x-1\right)^2\ge0\forall x\)
Do đó \(M\ge2014\forall x;y\)=> GTNN của M = 2014 khi \(\hept{\begin{cases}2x-1=0\\y+x+1=0\end{cases}\Leftrightarrow\hept{\begin{cases}x=\frac{1}{2}\\y=-\frac{3}{2}\end{cases}}}\).