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a) 5x2 + 10y2 - 6xy - 4x - 2y + 3
= ( x2 - 6xy + 9y2 ) + ( 4x2 - 4x + 1 ) + ( y2 - 2y + 1 ) + 1
= ( x - 3y )2 + ( 2x - 1 )2 + ( y - 1 )2 + 1
Ta có : \(\hept{\begin{cases}\left(x-3y\right)^2\\\left(2x-1\right)^2\\\left(y-1\right)^2\end{cases}}\ge0\forall x,y\Rightarrow\left(x-3y\right)^2+\left(2x-1\right)^2+\left(y-1\right)^2+1\ge1>0\forall x,y\)
=> đpcm
b) x2 + 4y2 + z2 - 2x - 6z + 8y + 15 = 0 < Sửa -z2 -> +z2 )
= ( x2 - 2x + 1 ) + ( 4y2 + 8y + 4 ) + ( z2 - 6z + 9 ) + 1
= ( x - 1 )2 + 4( y2 + 2y + 1 ) + ( z - 3 )2 + 1
= ( x - 1 )2 + 4( y + 1 )2 + ( z - 3 )2 + 1
Ta có : \(\hept{\begin{cases}\left(x-1\right)^2\ge0\forall x\\4\left(y+1\right)^2\ge0\forall y\\\left(z-3\right)^2\ge0\forall z\end{cases}}\Rightarrow\left(x-1\right)^2+4\left(y+1\right)^2+\left(z-3\right)^2+1\ge1>0\forall x,y,z\)
=> đpcm
Đặt \(\left(\frac{a-b}{c},\frac{b-c}{a},\frac{c-a}{b}\right)\rightarrow\left(x,y,z\right)\)
Khi đó:\(\left(\frac{c}{a-b},\frac{a}{b-c},\frac{b}{c-a}\right)\rightarrow\left(\frac{1}{x},\frac{1}{y},\frac{1}{z}\right)\)
Ta có:
\(P\cdot Q=\left(x+y+z\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=3+\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z}\)
Mặt khác:\(\frac{y+z}{x}=\left(\frac{b-c}{a}+\frac{c-a}{b}\right)\cdot\frac{c}{a-b}=\frac{b^2-bc+ac-a^2}{ab}\cdot\frac{c}{a-b}\)
\(=\frac{c\left(a-b\right)\left(c-a-b\right)}{ab\left(a-b\right)}=\frac{c\left(c-a-b\right)}{ab}=\frac{2c^2}{ab}\left(1\right)\)
Tương tự:\(\frac{x+z}{y}=\frac{2a^2}{bc}\left(2\right)\)
\(=\frac{x+y}{z}=\frac{2b^2}{ac}\left(3\right)\)
Từ ( 1 );( 2 );( 3 ) ta có:
\(P\cdot Q=3+\frac{2c^2}{ab}+\frac{2a^2}{bc}+\frac{2b^2}{ac}=3+\frac{2}{abc}\left(a^3+b^3+c^3\right)\)
Ta có:\(a+b+c=0\)
\(\Rightarrow\left(a+b\right)^3=-c^3\)
\(\Rightarrow a^3+b^3+3ab\left(a+b\right)=-c^3\)
\(\Rightarrow a^3+b^3+c^3=3abc\)
Khi đó:\(P\cdot Q=3+\frac{2}{abc}\cdot3abc=9\)
đề dài v~
1.
a) \(f\left(x\right)=5x^2-2x+1\)
\(5f\left(x\right)=25x^2-10x+5\)
\(5f\left(x\right)=\left(25x^2-10x+1\right)+4\)
\(5f\left(x\right)=\left(5x-1\right)^2+4\)
Mà \(\left(5x-1\right)^2\ge0\)
\(\Rightarrow5f\left(x\right)\ge4\)
\(\Leftrightarrow f\left(x\right)\ge\frac{4}{5}\)
Dấu " = " xảy ra khi :
\(5x-1=0\Leftrightarrow x=\frac{1}{5}\)
Vậy ....
b) \(P\left(x\right)=3x^2+x+7\)
\(3P\left(x\right)=9x^2+3x+21\)
\(3P\left(x\right)=\left(9x^2+3x+\frac{1}{4}\right)+\frac{83}{4}\)
\(3P\left(x\right)=\left(3x+\frac{1}{2}\right)^2+\frac{83}{4}\)
Mà \(\left(3x+\frac{1}{2}\right)^2\ge0\)
\(\Rightarrow3P\left(x\right)\ge\frac{83}{4}\)
\(\Leftrightarrow P\left(x\right)\ge\frac{83}{12}\)
Dấu "=" xảy ra khi :
\(3x+\frac{1}{2}=0\Leftrightarrow x=-\frac{1}{6}\)
Vậy ...
c) \(Q\left(x\right)=5x^2-3x-3\)
\(5Q\left(x\right)=25x^2-15x-15\)
\(\Leftrightarrow5Q\left(x\right)=\left(25x^2-15x+\frac{9}{4}\right)-\frac{69}{4}\)
\(\Leftrightarrow5Q\left(x\right)=\left(5x-\frac{3}{2}\right)^2-\frac{69}{4}\)
Mà \(\left(5x-\frac{3}{2}\right)^2\ge0\)
\(\Rightarrow5Q\left(x\right)\ge\frac{-69}{4}\)
\(\Leftrightarrow Q\left(x\right)\ge-\frac{69}{20}\)
Dấu "=" xảy ra khi :
\(5x-\frac{3}{2}=0\Leftrightarrow x=0,3\)
Vậy ...
2.
a) \(f\left(x\right)=-3x^2+x-2\)
\(-3f\left(x\right)=9x^2-3x+6\)
\(-3f\left(x\right)=\left(9x^2-3x+\frac{1}{4}\right)+\frac{23}{4}\)
\(-3f\left(x\right)=\left(3x-\frac{1}{2}\right)^2+\frac{23}{4}\)
Mà \(\left(3x-\frac{1}{2}\right)^2\ge0\)
\(\Rightarrow-3f\left(x\right)\ge\frac{23}{4}\)
\(\Leftrightarrow f\left(x\right)\le\frac{23}{12}\)
Dấu "=" xảy ra khi :
\(3x-\frac{1}{2}=0\Leftrightarrow x=\frac{1}{6}\)
Vậy ...
b) \(P\left(x\right)=-x^2-7x+1\)
\(-P\left(x\right)=x^2+7x-1\)
\(-P\left(x\right)=\left(x^2+7x+\frac{49}{4}\right)-\frac{53}{4}\)
\(-P\left(x\right)=\left(x+\frac{7}{2}\right)^2-\frac{53}{4}\)
Mà \(\left(x+\frac{7}{2}\right)^2\ge0\)
\(\Rightarrow-P\left(x\right)\ge-\frac{53}{4}\)
\(\Leftrightarrow P\left(x\right)\le\frac{53}{4}\)
Dấu "=" xảy ra khi :
\(x+\frac{7}{2}=0\Leftrightarrow x=-\frac{7}{2}\)
Vậy ...
c) \(Q\left(x\right)=-2x^2+x-8\)
\(-2Q\left(x\right)=4x^2-2x+16\)
\(-2Q\left(x\right)=\left(4x^2-2x+\frac{1}{4}\right)+\frac{63}{4}\)
\(-2Q\left(x\right)=\left(2x-\frac{1}{2}\right)^2+\frac{63}{4}\)
Mà : \(\left(2x-\frac{1}{2}\right)^2\ge0\)
\(\Rightarrow-2Q\left(x\right)\ge\frac{63}{4}\)
\(\Leftrightarrow Q\left(x\right)\le-\frac{63}{8}\)
Dấu "=" xảy ra khi :
\(2x-\frac{1}{2}=0\Leftrightarrow x=\frac{1}{4}\)
Vậy ...
1. Ta có : x + y + z = 0 \(\Rightarrow\)( x + y + z )2 = 0 \(\Rightarrow\)x2 + y2 + z2 = - 2 ( xy + yz + xz )\(S=\frac{x^2+y^2+z^2}{\left(y-z\right)^2+\left(z-x\right)^2+\left(x-y\right)^2}=\frac{-2\left(xy+yz+xz\right)}{2\left(x^2+y^2+z^2\right)-2\left(yz+xz+xy\right)}\)
\(S=\frac{-2\left(xy+yz+xz\right)}{-4\left(xy+yz+xz\right)-2\left(yz+xz+xy\right)}=\frac{-2\left(xy+yz+xz\right)}{-6\left(xy+yz+xz\right)}=\frac{1}{3}\)
3/ Ta có:
\(x+y+z=0\)
\(\Rightarrow x^2=\left(y+z\right)^2;y^2=\left(z+x\right)^2;z^2=\left(x+y\right)^2\)
\(a+b+c=0\)
\(\Rightarrow a+b=-c;b+c=-a;c+a=-b\)
\(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0\)
\(\Leftrightarrow ayz+bxz+cxy=0\)
Ta có:
\(ax^2+by^2+cz^2=a\left(y+z\right)^2+b\left(z+x\right)^2+c\left(x+y\right)^2\)
\(=x^2\left(b+c\right)+y^2\left(c+a\right)+z^2\left(a+b\right)+2\left(ayz+bzx+cxy\right)\)
\(=-ax^2-by^2-cz^2\)
\(\Leftrightarrow2\left(ax^2+by^2+cz^2\right)=0\)
\(\Leftrightarrow ax^2+by^2+cz^2=0\)
1/ Đặt \(a-b=x,b-c=y,c-z=z\)
\(\Rightarrow x+y+z=0\)
Ta có:
\(\frac{1}{\left(a-b\right)^2}+\frac{1}{\left(b-c\right)^2}+\frac{1}{\left(c-a\right)^2}=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\)
\(=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{2\left(x+y+z\right)}{xyz}\)
\(=\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+2\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\right)=\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2\)
Bài 2:
Ta có: \(a+b+c=2\)
\(\Leftrightarrow\left(a+b+c\right)^2=4\)
\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)=4\)
\(\Leftrightarrow2\left(ab+bc+ca\right)=2\)
\(\Rightarrow ab+bc+ca=1\)
Thay vào ta được: \(a^2+1=a^2+ab+bc+ca=\left(a+b\right)\left(a+c\right)\)
Tương tự CM được: \(b^2+1=\left(b+a\right)\left(b+c\right)\) và \(c^2+1=\left(c+a\right)\left(c+b\right)\)
=> \(M=\left(a^2+1\right)\left(b^2+1\right)\left(c^2+1\right)=\left[\left(a+b\right)\left(b+c\right)\left(c+a\right)\right]^2\)
=> đpcm