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1) Ta có : \(\frac{2016a+b+c+d}{a}=\frac{a+2016b+c+d}{b}=\frac{a+b+2016c+d}{c}=\frac{a+b+c+2016d}{d}\)
Trừ 4 vế với 2015 ta được : \(\frac{a+b+c+d}{a}=\frac{a+b+c+d}{b}=\frac{a+b+c+d}{c}=\frac{a+b+c+d}{d}\)
Nếu a + b + c + d = 0
=> a + b = -(c + d)
=> b + c = (-a + d)
=> c + d = -(a + b)
=> d + a = (-b + c)
Khi đó M = (-1) + (-1) + (-1) + (-1) = - 4
Nếu a + b + c + d\(\ne0\Rightarrow\frac{1}{a}=\frac{1}{b}=\frac{1}{c}=\frac{1}{d}\Rightarrow a=b=c=d\)
Khi đó M = 1 + 1 + 1 + 1 = 4
2) a) Ta có : \(\hept{\begin{cases}\left|x+2013\right|\ge0\forall x\\\left(3x-7\right)^{2004}\ge0\forall y\end{cases}\Rightarrow\left|x+2013\right|+\left(3x-7\right)^{2014}\ge0}\)
Dấu "=" xảy ra \(\hept{\begin{cases}x+2013=0\\3y-7=0\end{cases}\Rightarrow\hept{\begin{cases}x=-2013\\y=\frac{7}{3}\end{cases}}}\)
b) 72x + 72x + 3 = 344
=> 72x + 72x.73 = 344
=> 72x.(1 + 73) = 344
=> 72x = 1
=> 72x = 70
=> 2x = 0 => x = 0
c) Ta có :
\(\frac{7}{2x+2}=\frac{3}{2y-4}=\frac{5}{x+4}\Leftrightarrow\frac{7}{2x+2}=\frac{3}{2y-4}=\frac{10}{2x+8}=\frac{7-10}{2x+2-2x-8}=\frac{1}{2}\)(dãy tỉ số bằng nhau)
=> 2x + 2 = 14 => x = 6 ;
2y - 4 = 6 => y = 5 ;
6 + 5 + z = 17 => z = 6
Vậy x = 6 ; y = 5 ; z = 6
3) a) Ta có : \(\frac{a+b+c}{a+b-c}=\frac{a-b+c}{a-b-c}=\frac{a+b+c-a+b-c}{a+b-c-a+b+c}=\frac{2b}{2b}=1\)(dãy ti số bằng nhau)
=> a + b + c = a + b - c => a + b + c - a - b + c = 0 => 2c = 0 => c = 0;
Lại có : \(\frac{a+b+c}{a+b-c}-1=\frac{a-b+c}{a-b-c}-1\Leftrightarrow\frac{2c}{a+b-c}=\frac{2c}{a-b-c}\Rightarrow a+b-c=a-b-c\) => b = 0
Vậy c = 0 hoặc b = 0
c) Ta có : \(\frac{a+b}{c}=\frac{b+c}{a}=\frac{a+c}{b}=\frac{a+b+b+c+a+c}{c+a+b}=2\)(dãy tỉ số bằng nhau)
=> \(\hept{\begin{cases}a+b=2c\\b+c=2a\\a+c=2b\end{cases}}\)
Khi đó P = \(\left(1+\frac{c}{b}\right)\left(1+\frac{a}{c}\right)\left(1+\frac{b}{a}\right)=\frac{b+c}{b}.\frac{c+a}{c}=\frac{a+b}{a}=\frac{2a.2b.2c}{abc}=8\)
Vậy P = 8
2. b) \(7^{2x}+7^{2x+3}=344\)
\(7^{2x}\cdot\left(1+7^3\right)=344\)
\(7^{2x}\cdot\left(1+343\right)=344\)
\(7^{2x}\cdot344=344\)
\(7^{2x}=1\)
\(7^{2x}=7^0\)
\(2x=0\)
\(x=0\)
2.
a/\(A=5-I2x-1I\)
Ta thấy: \(I2x-1I\ge0,\forall x\)
nên\(5-I2x-1I\le5\)
\(A=5\)
\(\Leftrightarrow5-I2x-1I=5\)
\(\Leftrightarrow I2x-1I=0\)
\(\Leftrightarrow2x=1\)
\(\Leftrightarrow x=\frac{1}{2}\)
Vậy GTLN của \(A=5\Leftrightarrow x=\frac{1}{2}\)
b/\(B=\frac{1}{Ix-2I+3}\)
Ta thấy : \(Ix-2I\ge0,\forall x\)
nên \(Ix-2I+3\ge3,\forall x\)
\(\Rightarrow B=\frac{1}{Ix-2I+3}\le\frac{1}{3}\)
\(B=\frac{1}{3}\)
\(\Leftrightarrow B=\frac{1}{Ix-2I+3}=\frac{1}{3}\)
\(\Leftrightarrow Ix-2I+3=3\)
\(\Leftrightarrow Ix-2I=0\)
\(\Leftrightarrow x=2\)
Vậy GTLN của\(A=\frac{1}{3}\Leftrightarrow x=2\)
Áp dụng tính chất:\(|A|\ge0\)(Dấu "=" xảy ra khi và chỉ khi A=0)
Ta có\(A\ge0+0+0=0\)
Suy ra để A nhỏ nhát \(\Leftrightarrow\hept{\begin{cases}7x-5y=0\Rightarrow7x=5y\Rightarrow\frac{x}{5}=\frac{y}{7}\Rightarrow\frac{x}{10}=\frac{y}{14}\left(1\right)\\2z-3x=0\Rightarrow2z=3x\Rightarrow\frac{z}{3}=\frac{x}{2}\Rightarrow\frac{z}{15}=\frac{x}{10}\left(2\right)\\xy+yz+xz-2000=0\Rightarrow xy+yz+xz=2000\left(3\right)\end{cases}}\)
Từ (1) và (2)
\(\Rightarrow\frac{x}{10}=\frac{y}{14}=\frac{z}{15}=k\left(k\inℤ\right)\)
\(\Rightarrow\hept{\begin{cases}x=10k\\y=14k\\z=15k\end{cases}}\left(4\right)\)
Thay (4) vào (3)
\(\Rightarrow10k14k+14k15k+10k15k=2000\)
\(\Rightarrow140k^2+210k^2+150k^2=2000\)
\(\Rightarrow500k^2=2000\Rightarrow k^2=4=2^2=\left(-2\right)^2\)
Lần lượt thay K ta tìm đc các giá trị của x,y,z
Vậy ...
\(\frac{3x-2y}{37}=\frac{5y-3z}{15}=\frac{2z-5x}{2}=\)
\(\frac{3xz-2yz}{37z}=\frac{5yx-3zx}{15x}=\frac{2zy-5xy}{2y}=\frac{3xz-2yz+5yx-3zx+2zy-5xy}{37z+15x+2y}=0\)(t/c dãy tỉ số bằng nhau)
\(\frac{3x-2y}{37}=0\Rightarrow3x=2y\Rightarrow\frac{x}{2}=\frac{y}{3}\left(1\right)\)
\(\frac{5y-3z}{15}=0\Rightarrow5y=3z\Rightarrow\frac{z}{5}=\frac{y}{3}\left(2\right)\)
\(\frac{2z-5x}{2}=0\Rightarrow2z=5x\Rightarrow\frac{x}{2}=\frac{z}{5}\left(3\right)\)
\(\Rightarrow\frac{x}{2}=\frac{y}{3}=\frac{z}{5}=\frac{10x}{20}=\frac{3y}{9}=\frac{2z}{10}=\frac{10x-3y-2z}{20-9-10}=\frac{-4}{1}=-4\)
\(x=-8,y=-12,z=-20\)
Ta có: \(\frac{x}{y}=\frac{2}{3}\)
=> \(\frac{x}{2}=\frac{y}{3}\)=> \(\frac{x}{6}=\frac{y}{9}\)(1)
Có: \(\frac{x}{3}=\frac{z}{5}\)=> \(\frac{x}{6}=\frac{z}{10}\)(2)
Từ (1) ; (2) => \(\frac{x}{6}=\frac{y}{9}=\frac{z}{10}\)=> \(\frac{x^2}{36}=\frac{y^2}{81}=\frac{z^2}{100}=\frac{x^2+y^2+z^2}{36+81+100}=\frac{\frac{217}{4}}{217}=\frac{1}{4}\)
=> \(\hept{\begin{cases}\frac{x^2}{36}=\frac{1}{4}\\\frac{y^2}{81}=\frac{1}{4}\\\frac{z^2}{100}=\frac{1}{4}\end{cases}}\)=> \(\hept{\begin{cases}x^2=9\\y^2=\frac{81}{4}\\z^2=25\end{cases}}\)
Vì x, y, z dương nên suy ra: \(\hept{\begin{cases}x=3\\y=\frac{9}{2}\\z=5\end{cases}}\)
=> \(x+2y-2z=3+2.\frac{9}{2}-2.5=2\)
Ta có : \(\frac{x}{y}=\frac{2}{3};\frac{x}{3}=\frac{z}{5}\)
\(\Rightarrow\frac{x}{2}=\frac{y}{3};\frac{x}{3}=\frac{z}{5}\)
\(\Rightarrow\frac{x}{6}=\frac{y}{9};\frac{x}{6}=\frac{z}{10}\)
\(\Rightarrow\frac{x}{6}=\frac{y}{9}=\frac{z}{10}\)
Đặt \(\frac{x}{6}=\frac{y}{9}=\frac{z}{10}=k\)(k>0)
\(\Rightarrow\hept{\begin{cases}x=6k\\y=9k\\z=10k\end{cases}}\)
Thay x=6k; y=9k; z=10k vào \(x^2+y^2+z^2=\frac{217}{4}\) ta có:
\(\left(6k\right)^2+\left(9k\right)^2+\left(10k^2\right)=\frac{217}{4}\)
\(\Rightarrow6^2.k^2+9^2.k^2+10^2.k^2=\frac{217}{4}\)
\(\Rightarrow k^2.\left(6^2+9^2+10^2\right)=\frac{217}{4}\)
\(\Rightarrow k^2.\left(36+81+100\right)=\frac{217}{4}\)
\(\Rightarrow k^2.217=\frac{217}{4}\)
\(\Rightarrow k^2=\frac{217}{4}.\frac{1}{217}=\frac{1}{4}\)
\(\Rightarrow k=\pm\frac{1}{2}\)
Mà k >0
\(\Rightarrow k=\frac{1}{2}\)
\(\Rightarrow\hept{\begin{cases}x=6.\frac{1}{2}=3\\y=9.\frac{1}{2}=\frac{9}{2}\\z=10.\frac{1}{2}=5\end{cases}}\)( thỏa mãn x;y dương)
\(\Rightarrow x+2y-2z=3+2.\frac{9}{2}-2.5=3+9-10=2\)
Vậy x+2y-2z=2
1) \(2x=3y=5z\Leftrightarrow\frac{x}{15}=\frac{y}{10}=\frac{z}{6}=\frac{x-2y}{15-2\cdot10}=\frac{x-2y}{-5}\)
*TH1: Nếu x-2y = 5
\(\Rightarrow\frac{x}{15}=\frac{y}{10}=\frac{z}{6}=\frac{5}{-5}=-1\)
\(\Rightarrow\hept{\begin{cases}x=-15\\y=-10\\z=-6\end{cases}}\)\(\Rightarrow3x-2z=3\left(-15\right)-2\cdot6=-45-12=-57\)
*TH2: Nếu x-2y = -5
\(\Rightarrow\frac{x}{15}=\frac{y}{10}=\frac{z}{6}=1\)\(\Rightarrow\hept{\begin{cases}x=15\\y=10\\z=6\end{cases}\Rightarrow3x-2z=3\cdot15-2\cdot6=45-12=33}\)
Vậy giá trị nhỏ nhất của 3x - 2z là -57.
2)\(B=\frac{x^2+15}{x^2+3}=1+\frac{12}{x^2+3}\le1+\frac{12}{3}=5\)
Dấu "=" xảy ra khi x = 0.