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Ta có :
\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\Rightarrow\frac{1}{a}+\frac{1}{b}=-\frac{1}{c}\)
\(\Leftrightarrow\left(\frac{1}{a}+\frac{1}{b}\right)^3=\left(-\frac{1}{c}\right)^3\)
\(\Leftrightarrow\frac{1}{a^3}+\frac{1}{b^3}+3\frac{1}{a}.\frac{1}{b}\left(\frac{1}{a}+\frac{1}{b}\right)=-\frac{1}{c^3}\)
\(\Leftrightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=-3\frac{1}{a}\frac{1}{b}\left(\frac{1}{a}+\frac{1}{b}\right)\)
\(\Leftrightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=-3\frac{1}{a}\frac{1}{b}\left(-\frac{1}{c}\right)\)
\(\Leftrightarrow\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}=3\frac{1}{abc}=\frac{3}{abc}\)
Ta lại có :
\(P=\frac{bc}{a^2}+\frac{ca}{b^2}+\frac{ab}{c^2}=\frac{abc}{a^3}+\frac{bca}{b^3}+\frac{cab}{c^3}\)
\(=abc\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)=abc.\frac{3}{abc}=3\)
\(\)
Bài làm:
Ta có: \(P=\frac{bc}{a^2}+\frac{ca}{b^2}+\frac{ab}{c^2}=\frac{abc}{a^3}+\frac{abc}{b^3}+\frac{abc}{c^3}\)
\(=abc\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)\)
CM HĐT phụ:
Ta có: \(a^3+b^3+c^3=\left(a^3+b^3+c^3-3abc\right)+3abc\)
\(=\left[\left(a+b\right)^3-3ab\left(a+b\right)+c^3-3abc\right]+3abc\)
\(=\left[\left(a+b+c\right)\left(a^2+2ab+b^2-ac-bc+c^2\right)-3ab\left(a+b+c\right)\right]+3abc\)
\(=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ca\right)+3abc\)
Áp dụng vào trên ta được:
\(abc\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)\)
\(=abc\left[\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{1}{ab}-\frac{1}{bc}-\frac{1}{ca}\right)+\frac{3}{abc}\right]\)
Mà \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\)
\(P=abc.\frac{3}{abc}=3\)
Vậy P = 3
Bài làm:
Ta có: \(\left[\left(20-4x\right)\div\left(x^2-25\right)\right]+5\div\left(x+5\right)\)
\(=\frac{4\left(5-x\right)}{\left(x-5\right)\left(x+5\right)}+\frac{5}{x+5}\)
\(=\frac{-4}{x+5}+\frac{5}{x+5}\)
\(=\frac{1}{x+5}\)
\(\left[\left(20-4x\right):\left(x^2-25\right)\right]+\left[5:\left(x+5\right)\right]\)ĐK : x \(\ne\pm5\)
\(\Leftrightarrow\left[\frac{20-4x}{x^2-25}\right]+\left[\frac{5}{x+5}\right]\)
\(\Leftrightarrow\left[\frac{-4\left(x-5\right)}{\left(x-5\right)\left(x+5\right)}\right]+\left[\frac{5}{x+5}\right]\)
\(\Leftrightarrow\left[\frac{-4}{x+5}\right]+\left[\frac{5}{x+5}\right]=\frac{-4+5}{x+5}=\frac{1}{x+5}\)
gt <=> \(a^2+b^2+c^2-ab-bc-ca=0\)
<=> \(2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)
<=> \(\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)=0\)
<=> \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\) (1)
TA LUÔN CÓ: \(\left(a-b\right)^2;\left(b-c\right)^2;\left(c-a\right)^2\ge0\forall a;b;c\)
=> \(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\) (2)
TỪ (1) VÀ (2) => DẤU "=" SẼ XẢY RA <=> \(\left(a-b\right)^2=\left(b-c\right)^2=\left(c-a\right)^2=0\)
<=> \(a=b=c\)
VẬY TA CÓ ĐPCM.
a2 + b2 + c2 = ab + bc + ca
<=> 2( a2 + b2 + c2 ) = 2( ab + bc + ca )
<=> 2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca = 0
<=> ( a2 - 2ab + b2 ) + ( b2 - 2bc + c2 ) + ( c2 - 2ca + a2 ) = 0
<=> ( a - b )2 + ( b - c )2 + ( c - a )2 = 0 (*)
Ta có : \(\hept{\begin{cases}\left(a-b\right)^2\\\left(b-c\right)^2\\\left(c-a\right)^2\end{cases}}\ge0\forall a,b,c\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\)
Đẳng thức xảy ra ( tức là (*) xảy ra ) <=> \(\hept{\begin{cases}a-b=0\\b-c=0\\c-a=0\end{cases}}\Rightarrow a=b=c\)
=> ĐPCM
a)
\(A=\left(x^2-4x+4\right)+1=\left(x-2\right)^2+1\)
CÓ: \(\left(x-2\right)^2\ge0\forall x\Rightarrow\left(x-2\right)^2+1\ge1\)
=> \(A\ge1\)
DẤU "=" XẢY RA <=> \(x=2\)
b)
\(2B=4x^2+6x+2=\left(2x+\frac{3}{2}\right)^2-0,25\)
CÓ: \(\left(2x+\frac{3}{2}\right)^2\ge0\forall x\Rightarrow\left(2x+\frac{3}{2}\right)^2-0,25\ge-0,25\)
DẤU "=" XẢY RA <=> \(2x+\frac{3}{2}=0\Leftrightarrow x=-\frac{3}{4}\)
c)
\(C=\left(2x+\frac{5}{4}\right)^2-\frac{73}{16}\ge-\frac{73}{16}\)
DẤU "=" XẢY RA <=> \(2x+\frac{5}{4}=0\Leftrightarrow x=-\frac{5}{8}\)
a. Ta có :
\(A=x^2-4x+5=x^2-4x+4+1=\left(x-2\right)^2+1\)
Vì \(\left(x-2\right)^2\ge0\forall x\)\(\Rightarrow\left(x-2\right)^2+1\ge1\)
Dấu "=" xảy ra \(\Leftrightarrow\left(x-2\right)^2=0\Leftrightarrow x-2=0\Leftrightarrow x=2\)
b. \(B=2x^2+3x+1=2\left(x+\frac{3}{4}\right)^2-\frac{1}{8}\)
Vì \(\left(x+\frac{3}{4}\right)^2\ge0\forall x\)\(\Rightarrow2\left(x+\frac{3}{4}\right)^2-\frac{1}{8}\ge-\frac{1}{8}\)
Dấu "=" xảy ra \(\Leftrightarrow2\left(x+\frac{3}{4}\right)^2=0\Leftrightarrow x+\frac{3}{4}=0\Leftrightarrow x=-\frac{3}{4}\)
Vậy Bmin = - 1/8 <=> x = - 3/4
c. \(C=5x-3+4x^2=4\left(x+\frac{5}{8}\right)^2-\frac{73}{16}\)
Vì \(\left(x+\frac{5}{8}\right)^2\ge0\forall x\)\(\Rightarrow4\left(x+\frac{5}{8}\right)^2-\frac{73}{16}\ge-\frac{73}{16}\)
Dấu "=" xảy ra \(\Leftrightarrow4\left(x+\frac{5}{8}\right)^2=0\Leftrightarrow x+\frac{5}{8}=0\Leftrightarrow x=-\frac{5}{8}\)
Vậy Cmin = - 73/16 <=> x = - 5/8
CÓ: \(x^2+y^2=\left(x+y\right)^2-2xy=3^2-2.2=5\)
CÓ: \(x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)=3\left(5-2\right)=3.3=9\)
CÓ: \(x^4+y^4=\left(x^2+y^2\right)^2-2x^2y^2=5^2-2.2^2=25-8=17\)
CÓ: \(x^5+y^5=\left(x^4+y^4\right)\left(x+y\right)-x^4y-xy^4=3.17-xy\left(x^3+y^3\right)\)
\(=51-2.9=51-18=33\)
CÓ: \(x^6+y^6=\left(x+y\right)\left(x^5+y^5\right)-xy^5-x^5y\)
\(=3.33-xy\left(x^4+y^4\right)=3.33-2.17\)
\(=99-34=65\)
\(x^2+y^2=\left(x+y\right)^2-2xy=3^2-2.2=9-4=5\)
\(x^3+y^3=\left(x+y\right)^3-3xy\left(x+y\right)=3^3-3.2.3=27-18=9\)
\(x^4+y^4=\left(x+y\right)^4-4xy\left(x^2+y^2\right)-3xy.2xy\)
\(=3^4-4.2.5-3.2.2.2=81-40-24=17\)
Bài 11:
1) Sửa lại đề là: \(A=127^2+146.127+73^2\)
\(\Rightarrow A=127^2+2.127.73+73^2\)
\(\Rightarrow A=\left(127+73\right)^2\)
\(\Rightarrow A=200^2\)
\(\Rightarrow A=40000\)
Vậy \(A=40000.\)
2) Sửa lại đề là: \(B=9^8.2^8-\left(18^4-1\right).\left(18^4+1\right)\)
\(\Rightarrow B=\left(9.2\right)^8-\left[\left(18^4\right)^2-1^2\right]\)
\(\Rightarrow B=18^8-\left(18^8-1\right)\)
\(\Rightarrow B=18^8-18^8+1\)
\(\Rightarrow B=0+1\)
\(\Rightarrow B=1\)
Vậy \(B=1.\)
4) \(D=\left(3+1\right).\left(3^2+1\right).\left(3^4+1\right).\left(3^8+1\right).\left(3^{16}+1\right)\)
\(\Rightarrow2D=\left(3-1\right).\left(3+1\right).\left(3^2+1\right).\left(3^4+1\right).\left(3^8+1\right).\left(3^{16}+1\right)\)
\(=\left(3^2-1\right)\left(3^2+1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\)
\(=\left(3^4-1\right)\left(3^4+1\right)\left(3^8+1\right)\left(3^{16}+1\right)\)
\(=\left(3^8-1\right)\left(3^8+1\right)\left(3^{16}+1\right)\)
\(=\left(3^{16}-1\right)\left(3^{16}+1\right)\)
\(=3^{32}-1\)
\(\Rightarrow D=\frac{3^{32}-1}{2}\)