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\(T\ge\dfrac{\left(x+y+z\right)^2}{x+y+z+\sqrt{xy}+\sqrt{yz}+\sqrt{zx}}\ge\dfrac{\left(x+y+z\right)^2}{x+y+z+x+y+z}=\dfrac{x+y+z}{2}\ge\dfrac{2019}{2}\)
áp dụng BĐT:\(\dfrac{a^2}{x}+\dfrac{b^2}{y}+\dfrac{c^2}{z}\) với a,b,c,x,y,z là số dương
ta có BĐT Bunhiacopxki cho 3 bộ số:\(\left(\dfrac{a}{\sqrt{x}};\sqrt{x}\right);\left(\dfrac{b}{\sqrt{y}};\sqrt{y}\right);\left(\dfrac{c}{\sqrt{z}};\sqrt{z}\right)\)
ta có :
\(\dfrac{a^2}{x}+\dfrac{b^2}{y}+\dfrac{c^2}{z}\left(x+y+z\right)\)\(=\left[\left(\dfrac{a}{\sqrt{x}}\right)^2+\left(\dfrac{b}{\sqrt{y}}\right)^2+\left(\dfrac{c}{\sqrt{z}}\right)^2\right]\).\(\left[\left(\sqrt{x}\right)^2+\left(\sqrt{y}\right)^2+\left(\sqrt{z}\right)^2\right]\)\(\ge\left(\dfrac{a}{\sqrt{x}}.\sqrt{x}+\dfrac{b}{\sqrt{y}}.\sqrt{y}+\dfrac{c}{\sqrt{z}}.\sqrt{z}\right)^2=\left(a+b+c\right)^2\)
lúc đó ta có :\(\dfrac{a^2}{x}+\dfrac{b^2}{y}+\dfrac{c^2}{z}\ge\dfrac{\left(a+b+c\right)^2}{x+y+z}\)
ta có \(T=\dfrac{x^2}{x+\sqrt{yz}}+\dfrac{y^2}{y+\sqrt{zx}}+\dfrac{z^2}{z+\sqrt{xy}}\)\(\ge\dfrac{\left(x+y+z\right)^2}{x+\sqrt{yz}+y+\sqrt{zx}+z+\sqrt{xy}}\) mà ta có :
\(\sqrt{yz}+\sqrt{zx}+\sqrt{xy}\)\(\le\dfrac{x+y}{2}+\dfrac{x+z}{2}+\dfrac{z+y}{2}\)\(\Rightarrow\sqrt{yz}+\sqrt{zx}+\sqrt{xy}\le x+y+z\)
\(\Rightarrow T=\dfrac{2019}{2}\Leftrightarrow x=y=z=673\)
vậy \(\text{MinT}=\dfrac{2019}{2}\) khi và chỉ khi x=y=z=673
Ta có:\(P=\frac{x+y}{\sqrt{xy}}+\frac{\sqrt{xy}}{x+y}=\left(\frac{x+y}{4\sqrt{xy}}+\frac{\sqrt{xy}}{x+y}\right)+\frac{3\left(x+y\right)}{4\sqrt{xy}}\ge2\sqrt{\frac{x+y}{4\sqrt{xy}}.\frac{\sqrt{xy}}{x+y}}+\frac{3.2\sqrt{xy}}{4\sqrt{xy}}\)
\(=1+\frac{3}{2}=\frac{5}{2}\)
Dấu '=' xảy ra khi và chỉ khi x=y
Vậy P đạt GTNN của P là 5/2 khi x=y
-.-
\(x\ge xy+1\Rightarrow1\ge y+\dfrac{1}{x}\ge2\sqrt{\dfrac{y}{x}}\Rightarrow\dfrac{y}{x}\le\dfrac{1}{4}\)
\(Q^2=\dfrac{x^2+2xy+y^2}{3x^2-xy+y^2}=\dfrac{\left(\dfrac{y}{x}\right)^2+2\left(\dfrac{y}{x}\right)+1}{\left(\dfrac{y}{x}\right)^2-\dfrac{y}{x}+3}\)
Đặt \(\dfrac{y}{x}=t\le\dfrac{1}{4}\)
\(Q^2=\dfrac{t^2+2t+1}{t^2-t+3}=\dfrac{t^2+2t+1}{t^2-t+3}-\dfrac{5}{9}+\dfrac{5}{9}\)
\(Q^2=\dfrac{\left(4t-1\right)\left(t+6\right)}{9\left(t^2-t+3\right)}+\dfrac{5}{9}\le\dfrac{5}{9}\)
\(\Rightarrow Q_{max}=\dfrac{\sqrt{5}}{3}\) khi \(t=\dfrac{1}{4}\) hay \(\left(x;y\right)=\left(2;\dfrac{1}{2}\right)\)
\(3=x+y+xy\le\sqrt{2\left(x^2+y^2\right)}+\dfrac{x^2+y^2}{2}\)
\(\Rightarrow\left(\sqrt{x^2+y^2}-\sqrt{2}\right)\left(\sqrt{x^2+y^2}+3\sqrt{2}\right)\ge0\)
\(\Rightarrow x^2+y^2\ge2\)
\(\Rightarrow-\left(x^2+y^2\right)\le-2\)
\(P=\sqrt{9-x^2}+\sqrt{9-y^2}+\dfrac{x+y}{4}\le\sqrt{2\left(9-x^2+9-y^2\right)}+\dfrac{\sqrt{2\left(x^2+y^2\right)}}{4}\)
\(P\le\sqrt{2\left(18-x^2-y^2\right)}+\dfrac{1}{4}.\sqrt{2\left(x^2+y^2\right)}\)
\(P\le\left(\sqrt{2}-1\right)\sqrt{18-x^2-y^2}+\sqrt[]{2}\sqrt{\dfrac{\left(18-x^2-y^2\right)}{2}}+\dfrac{1}{2}\sqrt{\dfrac{x^2+y^2}{2}}\)
\(P\le\left(\sqrt{2}-1\right).\sqrt{18-2}+\sqrt{\left(2+\dfrac{1}{4}\right)\left(\dfrac{18-x^2-y^2+x^2+y^2}{2}\right)}=\dfrac{1+8\sqrt{2}}{2}\)
Dấu "=" xảy ra khi \(x=y=1\)
Ta có x2-xy+y2=\(\left(\dfrac{x+y}{2}\right)^2+3\left(\dfrac{x-y}{2}\right)^2\)\(\ge\)\(\left(\dfrac{x+y}{2}\right)^2\)
=>\(\dfrac{\sqrt{x^2-xy+y^2}}{x+y+2z}\ge\dfrac{x+y}{2\left(x+y+2z\right)}\)(1) . Tương tự ...
Đặt \(\left\{{}\begin{matrix}y+z=a\\x+z=b\\x+y=c\end{matrix}\right.\)(a,b,c>0). Khi đó ta có :
S=\(\dfrac{1}{2}\left(\dfrac{c}{a+b}+\dfrac{b}{a+c}+\dfrac{a}{b+c}\right)\ge\dfrac{3}{4}\) (Netbit)
Đặt \(\left(x;2y;3z\right)=\left(a;b;c\right)\Rightarrow a+b+c=2\)
\(S=\sqrt{\dfrac{ab}{ab+2c}}+\sqrt{\dfrac{bc}{bc+2a}}+\sqrt{\dfrac{ca}{ca+2b}}\)
\(S=\sqrt{\dfrac{ab}{ab+c\left(a+b+c\right)}}+\sqrt{\dfrac{bc}{bc+a\left(a+b+c\right)}}+\sqrt{\dfrac{ca}{ca+b\left(a+b+c\right)}}\)
\(S=\sqrt{\dfrac{ab}{\left(a+c\right)\left(b+c\right)}}+\sqrt{\dfrac{bc}{\left(a+b\right)\left(a+c\right)}}+\sqrt{\dfrac{ca}{\left(a+b\right)\left(b+c\right)}}\)
\(S\le\dfrac{1}{2}\left(\dfrac{a}{a+c}+\dfrac{b}{b+c}+\dfrac{b}{a+b}+\dfrac{c}{a+c}+\dfrac{a}{a+b}+\dfrac{c}{b+c}\right)=\dfrac{3}{2}\)
Dấu "=" xảy ra khi \(a=b=c=\dfrac{2}{3}\Rightarrow x;y;z\)
Đặt `(x+y)/sqrt{xy}=a(a>0)`
`P=a+1/a`
`=a+4/a-3/a`
Áp dụng BĐT cosi:
`a+4/a>=4`
`x+y>=2sqrt{xy}<=>sqrt{xy}/(x+y)<=1/2`
`<=>1/a<=1/2`
`<=>3/a<=3/2`
`<=>P>=4-3/2=8/2`
Dấu "=" `<=>x=y=1.`
???? hơi lỗi