1Cho x,y,z thỏa mãn $$x+y+z=1$$TÌM $Min$ của
$$M=\sqrt{x^{2}+xy+y^{2}}+\sqrt{y^{2}+yz+z^{2}}+\sqrt{x^{2}+xz+z^{2}}$$
2
Cho $a+b+c=0$ tính
$$P=(\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b})(\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a})$$
ta dễ dàng chứng minh được $\sqrt{x^2+xy+y^2} \geq  \frac{\sqrt{3}}{2}(x+y)$
$TH1 x+y<0 =>$ BĐT luôn đúng
$TH2x+y\geq 0=>$ Bình phương 2 vế $ \Leftrightarrow x^2+xy+y^2\geq \frac34(x^2+2xy+y^2)$
                                                                 $\Leftrightarrow 4x^2+4xy+4y^2\geq 3x^2+6xy+3y^2$
                                                                 $\Leftrightarrow x^2-2xy+y^2$
                                                                 $ \Leftrightarrow (x-y)^2 \geqslant 0 $
                   $ \Rightarrow M \geq  \frac{\sqrt{3}.2(x+y+z)}{2}=\sqrt{3}$
dấu $=$ xẩy ra $\Leftrightarrow x=y=z=\frac{1}{3}$


Bài 1 Ta có $2M=\sqrt{(2x+y)^2+(\sqrt3y)^2}+\sqrt{(2y+z)^2+(\sqrt3z)^2}+\sqrt{(2z+x)^2+(\sqrt3x)^2}$
Áp dụng BĐT Minkowxky 
=> $2M\geq \sqrt{(3x+3y+3z)^2+(\sqrt3x+\sqrt3y+\sqrt3z)^2}=\sqrt{9(x+y+z)^2+3(x+y+z)^2}=\sqrt{12}$
=> $M\geq \sqrt3$
Dấu = xảy ra tại$x=y=z=\frac13$
Vậy $minM\sqrt3$ tại $ x=y=z=\frac13$
bài 2
$A=\frac{a-b}{c}+\frac{b-c}{a}+\frac{c-a}{b}=\frac{a^2-ab^2+b^2c-bc^2+c^2a-ca^2}{abc}$
PTĐT thành nhân tử => $A=\frac{(a-b)(a-c)(b-c)}{abc}$
$B=\frac c{a-b}+\frac a{b-c}+\frac c{c-a}=\frac{bc^2-abc-c^3+ac^2+a^2c-a^3-abc+a^2b+ab^2-b^3-abc+b^2}{(a-b)(b-c)(c-a)}=\frac{a^3+b^3+c^3-a^2b-ab^2-b^2c-bc^2-a^2c-ac^2+3abc}{(a-b)(a-c)(b-c)}$
Tử thức $=(a+b)^3+c^3-4a^2b-4ab^2-b^c-bc^2-a^c-ac^2+3abc=(a+b+c)((a+b)^2-(a+b)c+c^2)-4ab(a+b)-bc(b+c)-ac(a+c)+3abc$
Vì $a+b+c=0$=>$\begin{cases}a+b=-c \\a+c=-b\\ b+c=-a \end{cases}$
=>$B=\frac{9abc}{(a-b)(a-c)(b-c)}$
=> $P=A.B=9$

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