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7. \(S=9y^2-12\left(x+4\right)y+\left(5x^2+24x+2016\right)\)
\(=9y^2-12\left(x+4\right)y+4\left(x+4\right)^2+\left(x^2+8x+16\right)+1936\)
\(=\left[3y-2\left(x+4\right)\right]^2+\left(x-4\right)^2+1936\ge1936\)
Vậy \(S_{min}=1936\) \(\Leftrightarrow\) \(\hept{\begin{cases}3y-2\left(x+4\right)=0\\x-4=0\end{cases}}\) \(\Leftrightarrow\) \(\hept{\begin{cases}x=4\\y=\frac{16}{3}\end{cases}}\)
7. \(S=9y^2-12\left(x+4\right)y+\left(5x^2+24x+2016\right)\)
\(=9y^2-12\left(x+4\right)y+4\left(x+4\right)^2+\left(x^2+8x+16\right)+1936\)
\(=\left[3y-2\left(x+4\right)\right]^2+\left(x-4\right)^2+1936\ge1936\)
Vậy \(S_{min}=1936\) \(\Leftrightarrow\) \(\hept{\begin{cases}3y-2\left(x+4\right)=0\\x-4=0\end{cases}}\) \(\Leftrightarrow\) \(\hept{\begin{cases}x=4\\y=\frac{16}{3}\end{cases}}\)
8. \(x^2-5x+14-4\sqrt{x+1}=0\) (ĐK: x > = -1).
\(\Leftrightarrow\) \(\left(x+1\right)-4\sqrt{x+1}+4+\left(x^2-6x+9\right)=0\)
\(\Leftrightarrow\) \(\left(\sqrt{x+1}-2\right)^2+\left(x-3\right)^2=0\)
Với mọi x thực ta luôn có: \(\left(\sqrt{x+1}-2\right)^2\ge0\) và \(\left(x-3\right)^2\ge0\)
Suy ra \(\left(\sqrt{x+1}-2\right)^2+\left(x-3\right)^2\ge0\)
Đẳng thức xảy ra \(\Leftrightarrow\) \(\hept{\begin{cases}\left(\sqrt{x+1}-2\right)^2=0\\\left(x-3\right)^2=0\end{cases}}\) \(\Leftrightarrow\) x = 3 (Nhận)
c) Có \(P=\frac{ax+b}{x^2+1}=-1+\frac{x^2+ax+b+1}{x^2+1}\);
\(P=\frac{ax+b}{x^2+1}=4-\frac{4x^2-ax-b+4}{x^2+1}\)
Để Min P = 1 và Max P = 4 thì
\(\hept{\begin{cases}x^2+ax+b+1=\left(x+c\right)^2\\4x^2-ax-b+4=\left(2x+d\right)^2\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x\left(a-2c\right)+\left(b+1-c^2\right)=0\left(1\right)\\x\left(-a-4d\right)+\left(-b+4-d^2\right)=0\left(2\right)\end{cases}}\)
(1) = 0 khi \(\hept{\begin{cases}a=2c\\b=c^2-1\end{cases}}\)(3)
(2) = 0 khi \(\hept{\begin{cases}a=-4d\\b=4-d^2\end{cases}}\)(4)
Từ (3) (4) => d = 1 ; c = -2 ; b = 3 ; a = -4
Vậy \(P=\frac{-4x+3}{x^2+1}\)
ĐK \(x\ge y\)
Đặt \(\sqrt{x+y}=a;\sqrt{x-y}=b\left(a;b\ge0\right)\)
HPT <=> \(\hept{\begin{cases}a^4+b^4=82\\a-2b=1\end{cases}}\Leftrightarrow\hept{\begin{cases}\left(2b+1\right)^4+b^4=82\\a=2b+1\end{cases}}\Leftrightarrow\hept{\begin{cases}17b^4+32b^3+24b^2+8b-81=0\\a=2b+1\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}17b^4-17b^3+49^3-49b^2+73b^2-73b+81b-81=0\\a=2b+1\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}\left(b-1\right)\left(17b^3+49b^2+73b+81\right)=0\left(1\right)\\a=2b+1\end{cases}}\)
Giải (1) ; kết hợp điều kiện => b = 1
=> Hệ lúc đó trở thành \(\hept{\begin{cases}b=1\\a=2b+1\end{cases}}\Leftrightarrow\hept{\begin{cases}b=1\\a=3\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}\sqrt{x+y}=3\\\sqrt{x-y}=1\end{cases}}\Leftrightarrow\hept{\begin{cases}x+y=9\\x-y=1\end{cases}}\Leftrightarrow\hept{\begin{cases}2x=10\\x-y=1\end{cases}}\Leftrightarrow\hept{\begin{cases}x=5\\x-y=1\end{cases}}\Leftrightarrow\hept{\begin{cases}x=5\\y=4\end{cases}}\)
Vậy hệ có 1 nghiệm duy nhất (x;y) = (5;4)
Đặt \(\left(\frac{1}{a},\frac{1}{b},\frac{1}{c}\right)=\left(x,y,z\right)\)
\(x+y+z\ge\frac{x^2+2xy}{2x+y}+\frac{y^2+2yz}{2y+z}+\frac{z^2+2zx}{2z+x}\)
\(\Leftrightarrow x+y+z\ge\frac{3xy}{2x+y}+\frac{3yz}{2y+z}+\frac{3zx}{2z+x}\)
\(\frac{3xy}{2x+y}\le\frac{3}{9}xy\left(\frac{1}{x}+\frac{1}{x}+\frac{1}{y}\right)=\frac{1}{3}\left(x+2y\right)\)
\(\Rightarrow\Sigma_{cyc}\frac{3xy}{2x+y}\le\frac{1}{3}\left[\left(x+2y\right)+\left(y+2z\right)+\left(z+2x\right)\right]=x+y+z\)
Dấu "=" xảy ra khi x=y=z
Ta có:\(7\left(\frac{1}{a^2}+...\right)=6\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)+2015\)
Mà \(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\le\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\)
=> \(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\le2015\)=> \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le\sqrt{6045}\)
\(P=\frac{1}{\sqrt{3\left(2a^2+b^2\right)}}+...\)
Mà \(\left(2+1\right)\left(2a^2+b^2\right)\ge\left(2a+b\right)^2\)(bất dẳng thức buniacoxki)
=> \(P\le\frac{1}{2a+b}+\frac{1}{2b+c}+\frac{1}{2c+a}\)
Lại có \(\frac{1}{2a+b}=\frac{1}{a+a+b}\le\frac{1}{9}\left(\frac{1}{a}+\frac{1}{a}+\frac{1}{b}\right)\)
=> \(P\le\frac{1}{9}\left(\frac{3}{a}+\frac{3}{b}+\frac{3}{c}\right)\le\frac{\sqrt{6045}}{3}\)
Vậy \(MaxP=\frac{\sqrt{6045}}{3}\)khi \(a=b=c=\frac{\sqrt{6045}}{2015}\)
\(\frac{b\left(2a-b\right)}{a\left(b+c\right)}+\frac{c\left(2b-c\right)}{b\left(c+a\right)}+\frac{a\left(2c-a\right)}{c\left(a+b\right)}\le\frac{3}{2}\)
\(\Leftrightarrow\left[2-\frac{b\left(2a-b\right)}{a\left(b+c\right)}\right]+\left[2-\frac{c\left(2b-c\right)}{b\left(c+a\right)}\right]+\left[2-\frac{a\left(2c-a\right)}{c\left(a+b\right)}\right]\ge\frac{9}{2}\)
\(\Leftrightarrow\frac{b^2+2ca}{a\left(b+c\right)}+\frac{c^2+2ab}{b\left(c+a\right)}+\frac{a^2+2bc}{c\left(a+b\right)}\ge\frac{9}{2}\)
Áp dụng BĐT Schwarz, ta có :
\(\frac{b^2}{a\left(b+c\right)}+\frac{c^2}{b\left(c+a\right)}+\frac{a^2}{c\left(a+b\right)}\ge\frac{\left(a+b+c\right)^2}{a\left(b+c\right)+b\left(c+a\right)+c\left(a+b\right)}=\frac{\left(a+b+c\right)^2}{2\left(ab+bc+ac\right)}\)( 1 )
\(\frac{ac}{a\left(b+c\right)}+\frac{ab}{b\left(c+a\right)}+\frac{bc}{c\left(a+b\right)}=\frac{c^2}{c\left(b+c\right)}+\frac{a^2}{a\left(a+c\right)}+\frac{b^2}{b\left(a+b\right)}\) ( 2 )
\(\ge\frac{\left(a+b+c\right)^2}{a^2+b^2+c^2+ab+bc+ac}\)
Cộng ( 1 ) với ( 2 ), ta được :
\(\frac{b^2+2ca}{a\left(b+c\right)}+\frac{c^2+2ab}{b\left(c+a\right)}+\frac{a^2+2bc}{c\left(a+b\right)}\)
\(\ge\left(a+b+c\right)^2\left(\frac{1}{2\left(ab+bc+ac\right)}+\frac{2}{a^2+b^2+c^2+ab+bc+ac}\right)\)
\(\ge\left(a+b+c\right)^2\left(\frac{\left(1+2\right)^2}{2\left(ab+bc+ac\right)+2\left(a^2+b^2+c^2+ab+bc+ac\right)}\right)=\frac{9}{2}\)
không biết cách này ổn không
Ta có : \(\frac{b\left(2a-b\right)}{a\left(b+c\right)}=\frac{2-\frac{b}{a}}{\frac{c}{b}+1}\) ; tương tự :...
đặt \(\frac{a}{c}=x;\frac{b}{a}=y;\frac{c}{b}=z\Rightarrow xyz=1\)
\(\Sigma\frac{2-y}{z+1}\le\frac{3}{2}\)
\(\Leftrightarrow2\Sigma xy^2+2\Sigma x^2+\Sigma xy\ge3\Sigma x+6\)( quy đồng khử mẫu )
\(\Leftrightarrow\Sigma\frac{x}{y}\ge\Sigma x\)( xyz = 1 ) ( luôn đúng )
\(\Rightarrowđpcm\)