1Cho x,y,z thỏa mãn x+y+z=1TÌM Min của
M=x2+xy+y2+y2+yz+z2+x2+xz+z2
2
Cho a+b+c=0 tính
P=(abc+bca+cab)(cab+abc+bca)
ta dễ dàng chứng minh được x2+xy+y232(x+y)
TH1x+y<0=> BĐT luôn đúng
TH2x+y0=> Bình phương 2 vế x2+xy+y234(x2+2xy+y2)
                                                                 4x2+4xy+4y23x2+6xy+3y2
                                                                 x22xy+y2
                                                                 (xy)2
                    \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 tạix=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
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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