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4. Ta có: \(a+b+c=6abc\)
\(\Rightarrow\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}=6\)
Đặt \(\frac{1}{a}=x;\frac{1}{b}=y;\frac{1}{c}=z\)
\(\Rightarrow xy+yz+zx=6\)
Lại có: \(\frac{bc}{a^3\left(c+2b\right)}=\frac{1}{a^3\frac{c+2b}{bc}}=\frac{\frac{1}{a^3}}{\frac{1}{b}+\frac{2}{c}}=\frac{x^3}{y+2z}\)
Tương tự suy ra:
\(S=\frac{x^3}{y+2z}+\frac{y^3}{z+2x}+\frac{z^3}{x+2y}\)
\(=\frac{x^4}{xy+2zx}+\frac{y^4}{yz+2xy}+\frac{z^4}{zx+2yz}\)
\(\ge\frac{\left(x^2+y^2+z^2\right)^2}{3\left(xy+yz+zx\right)}\ge\frac{x^2+y^2+z^2}{3}\ge\frac{xy+yz+zx}{3}=2\)
Dấu = xảy ra khi \(x=y=z=\sqrt{2}\Rightarrow a=b=c=\frac{1}{\sqrt{2}}\)
1, với x > 0 ; x khác 1 ; 4
a, \(P=\left(\dfrac{x+\sqrt{x}-x-2}{\sqrt{x}+1}\right):\left(\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)+\sqrt{x}-4}{x-1}\right)\)
\(=\dfrac{\sqrt{x}-2}{\sqrt{x}+1}:\dfrac{x-4}{x-1}=\dfrac{\sqrt{x}-1}{\sqrt{x}+2}\)
b, Ta có P > 0 => \(\sqrt{x}-1>0\Leftrightarrow x>1\)
Kết hợp đk vậy x > 1 ; x khác 4
1) \(\left(a;b\right)=\left(\sqrt{3x+4y};\sqrt{8-x+y}\right)\) \(\left(a;b\ge0\right)\)
hpt \(\Leftrightarrow\)\(\hept{\begin{cases}4a+b=23\\3b-2\sqrt{-a^2-9b^2+110}=5\end{cases}}\Leftrightarrow\hept{\begin{cases}b=23-4a\\32-6a=\sqrt{-145a^2+1656a-4651}\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}b=23-4a\\181a^2-2040a+5675=0\left(1\right)\end{cases}}\)
(1) \(\Leftrightarrow\)\(\orbr{\begin{cases}a=5\left(nhan\right)\Rightarrow b=3\left(nhan\right)\\a=\frac{1135}{181}\left(nhan\right)\Rightarrow b=\frac{-377}{181}\left(loai\right)\end{cases}}\)\(\Rightarrow\)\(a=5;b=3\)\(\Rightarrow\)\(x=3;y=4\)
Chuẩn hóa \(a+b+c=3\)
WLOG \(a\le b\le c\)
Ta có:
\(\left(a+b+c\right)\left(a^2+b^2+c^2\right)-3\left(ab^2+bc^2+ca^2\right)=\left(a+b\right)\left(a-b\right)^2+\left(2a-b+c\right)\left(c-a\right)\left(c-b\right)\ge0\)
\(\Sigma_{cyc}a.\Sigma_{cyc}a^2\ge3\Sigma_{cyc}ab^2\)
\(ab^2+bc^2+ca^2-a^2b-b^2c-c^2a=\left(a-b\right)\left(b-c\right)\left(c-a\right)\ge0\)
\(\Sigma_{cyc}ab^2\ge\Sigma_{cyc}a^2b\)
Giờ ta áp dụng hai bđt trên:
\(\Sigma_{cyc}\frac{a^2}{b}\ge\frac{\left(a^2+b^2+c^2\right)^2}{a^2b+b^2c+c^2a}\ge\frac{\left(a^2+b^2+c^2\right)^2}{ab^2+bc^2+ca^2}\ge\frac{3\left(a^2+b^2+c^2\right)}{a+b+c}=a^2+b^2+c^2\left(\cdot\right)\)
\(\hept{\begin{cases}\sqrt{\frac{a^2+b^2}{2}}\le\frac{a^2+b^2+2}{4}\\\sqrt{\frac{b^2+c^2}{2}}\le\frac{b^2+c^2+2}{4}\\\sqrt{\frac{c^2+a^2}{2}}\le\frac{c^2+a^2+2}{4}\end{cases}\Rightarrow\Sigma_{cyc}\sqrt{\frac{a^2+b^2}{2}}\le\frac{1}{2}\left(a^2+b^2+c^2\right)+\frac{3}{2}\left(\cdot\cdot\right)}\)
Với:
\(a^2+b^2+c^2\ge3\Rightarrow a^2+b^2+c^2\ge\frac{1}{2}\left(a^2+b^2+c^2\right)+\frac{3}{2}\left(\cdot\cdot\cdot\right)\) \(\left(\cdot\right),\left(\cdot\cdot\cdot\right)và\left(\cdot\cdot\cdot\right)\Rightarrow\Sigma_{cyc}\frac{a^2}{b}\ge\Sigma_{cyc}\sqrt{\frac{a^2+b^2}{2}}\)
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)
Mới nghĩ ra 3 câu:
a/ \(\frac{ab}{\sqrt{\left(1-c\right)^2\left(1+c\right)}}=\frac{ab}{\sqrt{\left(a+b\right)^2\left(1+c\right)}}\le\frac{ab}{2\sqrt{ab\left(1+c\right)}}=\frac{1}{2}\sqrt{\frac{ab}{1+c}}\)
\(\sum\sqrt{\frac{ab}{1+c}}\le\sqrt{2\sum\frac{ab}{1+c}}\)
\(\sum\frac{ab}{1+c}=\sum\frac{ab}{a+c+b+c}\le\frac{1}{4}\sum\left(\frac{ab}{a+c}+\frac{ab}{b+c}\right)=\frac{1}{4}\)
c/ \(ab+bc+ca=2abc\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=2\)
Đặt \(\left(x;y;z\right)=\left(\frac{1}{a};\frac{1}{b};\frac{1}{c}\right)\Rightarrow x+y+z=2\)
\(VT=\sum\frac{x^3}{\left(2-x\right)^2}\)
Ta có đánh giá: \(\frac{x^3}{\left(2-x\right)^2}\ge x-\frac{1}{2}\) \(\forall x\in\left(0;2\right)\)
\(\Leftrightarrow2x^3\ge\left(2x-1\right)\left(x^2-4x+4\right)\)
\(\Leftrightarrow9x^2-12x+4\ge0\Leftrightarrow\left(3x-2\right)^2\ge0\)
d/ Ta có đánh giá: \(\frac{x^4+y^4}{x^3+y^3}\ge\frac{x+y}{2}\)
\(\Leftrightarrow\left(x-y\right)^2\left(x^2+xy+y^2\right)\ge0\)
Akai Haruma, Nguyễn Ngọc Lộc , @tth_new, @Băng Băng 2k6, @Trần Thanh Phương, @Nguyễn Việt Lâm
Mn giúp e vs ạ! Thanks!
