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vào link này tham khảo nha http://olm.vn/hoi-dap/question/461515.html
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Ta có :
\(K=\left(-x^2-9y^2-1+6xy+6y-2x\right)+\left(-y^2+4y-4\right)+2015\)
\(=-\left[x^2+\left(3y\right)^2+1^2+2.x.3y+2.x.\left(-1\right)+2.3y.1\right]-\left(y^2-4y+4\right)+2015\)
\(=-\left(x-3y+1\right)^2-\left(y-2\right)^2+2015\)
Ta thấy \(-\left(x-3y+1\right)^2\le0\forall x;y\text{ }\text{and}\text{ }-\left(y-2\right)^2\le0\forall y\)
\(\Rightarrow-\left(x-3y+1\right)^2-\left(y-2\right)^2\le0\forall x;y\)
\(\Rightarrow K=-\left(x-3y+1\right)^2-\left(y-2\right)^2+2015\le2015\forall x;y\)
K đạt GTLN là 2015 khi \(\hept{\begin{cases}x-3y+1=0\\y-2=0\end{cases}}\Rightarrow\hept{\begin{cases}x=5\\y=2\end{cases}}\)
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Ta có: \(xyz\le\left(\frac{x+y+z}{3}\right)^3=\frac{1}{27}\) và \(\left(x+y\right)\left(y+z\right)\left(z+x\right)\le\left(\frac{x+y+y+z+z+x}{3}\right)^3=\frac{8}{27}\)
\(\Rightarrow B\le\frac{1}{27}.\frac{8}{27}=\frac{8}{729}\Rightarrow k=\frac{8}{729}\Rightarrow9^3.k=8\)
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\(S=5x^2+y^2+2.\left(\sqrt{2}x\right)\left(\frac{y}{\sqrt{2}}\right)\le5x^2+y^2+2x^2+\frac{y^2}{2}\)
\(\Rightarrow S\le7x^2+\frac{3}{2}y^2=\frac{1}{2}\left(14x^2+3y^2\right)=\frac{2019}{2}\)
Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}14x^2+3y^2=2019\\y=2x\end{matrix}\right.\) \(\Rightarrow...\)
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a) \(A=\left(\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}+\dfrac{1}{1-\sqrt{x}}\right):\dfrac{\sqrt{x}-1}{2}\) (ĐK: \(x\ne1,x\ge0\))
\(A=\left[\dfrac{x+2}{\left(\sqrt{x}\right)^3-1^3}+\dfrac{\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\right]\cdot\dfrac{2}{\sqrt{x}-1}\)
\(A=\left[\dfrac{\left(x+2\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\dfrac{x+\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\right]\cdot\dfrac{2}{\sqrt{x}-1}\)
\(A=\dfrac{x+2+x-\sqrt{x}-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\cdot\dfrac{2}{\sqrt{x}-1}\)
\(A=\dfrac{x-2\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\cdot\dfrac{2}{\sqrt{x}-1}\)
\(A=\dfrac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\cdot\dfrac{2}{\sqrt{x}-1}\)
\(A=\dfrac{\sqrt{x}-1}{x+\sqrt{x}+1}\cdot\dfrac{2}{\sqrt{x}-1}\)
\(A=\dfrac{2}{x+\sqrt{x}+1}\)
b) Ta có:
\(A=\dfrac{2}{x+\sqrt{x}+1}=\dfrac{2}{x+2\cdot\dfrac{1}{2}\cdot x+\dfrac{1}{4}+\dfrac{3}{4}}=\dfrac{2}{\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}}\)
Mà: \(2>0\Rightarrow\dfrac{2}{\left(x+\dfrac{1}{2}\right)+\dfrac{3}{4}}\le\dfrac{2}{\dfrac{3}{4}}=\dfrac{8}{3}\)
Dấu "=" xảy ra:
\(\dfrac{2}{\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}}=\dfrac{8}{3}\)
\(\Leftrightarrow\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}=2:\dfrac{8}{3}\)
\(\Leftrightarrow\left(x+\dfrac{1}{2}\right)^2+\dfrac{3}{4}=\dfrac{3}{4}\Leftrightarrow x+\dfrac{1}{2}=0\Leftrightarrow x=-\dfrac{1}{2}\)
Vậy: \(A_{max}=\dfrac{8}{3}\) khi \(x=-\dfrac{1}{2}\)
\(A=-\left(x^2+y^2+3^2+2xy-6x-6y\right)-4\left(x^2-2x+1\right)-\left(y^2-4y+4\right)-3\)
\(A=-\left(x+y-3\right)^2-4\left(x-1\right)^2-\left(y-2\right)^2-3\le-3\)
Vậy Max A=-3 <=> x=1;y=2