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Đặt \(\left\{{}\begin{matrix}\sqrt{x}=a\\\sqrt{y}=b\end{matrix}\right.\), ta có:
\(A=\left[\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\times\dfrac{2}{a+b}+\dfrac{1}{a^2}+\dfrac{1}{b^2}\right]\)\(\times\dfrac{a^3+ab^2+a^2b+b^3}{ab^3+a^3b}\)
\(=\left(\dfrac{b+a}{ab}\times\dfrac{2}{a+b}+\dfrac{b^2+a^2}{a^2b^2}\right)\)\(\times\dfrac{a^2\left(a+b\right)+b^2\left(a+b\right)}{ab\left(a^2+b^2\right)}\)
\(=\dfrac{2ab+b^2+a^2}{a^2b^2}\times\dfrac{\left(a+b\right)\left(a^2+b^2\right)}{ab\left(b^2+a^2\right)}\)
\(=\dfrac{\left(a+b\right)^3}{a^3b^3}\)
\(=\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^3}{\sqrt{\left(xy\right)^3}}\)
Bổ sung giả thuyết x ,y \(\ge0\)
Do giả thiết x ,y \(\ge0\)
\(\sqrt{x}+\sqrt{y}\) =1
nên:
xy (x+y )\(^2\)\(\le\) \(\dfrac{1}{64}\)
<=> 64 xy (x + y )\(^2\) \(\le\)1
<=> 64 xy ( x + y)\(^2\)\(\le\)(\(\sqrt{x}+\sqrt{y}\))\(^8\)
<=> 64 xy ( x + y )\(^2\) < \((x+2\sqrt{xy}+y)^4\)
Áp dụng bất đẳng thức Cauchy cho 2 số không âm x + y và \(2\sqrt{xy}\)
ta có ;
x + y + 2\(\sqrt{xy}\) \(\ge\) \(2\sqrt{x+y}2\sqrt{xy}\)
=> ( x + y +2\(\sqrt{xy}\)) \(^4\)\(\ge\) (\(2\sqrt{x+y}2\sqrt{xy}\) )\(^4\)= 64 xy (x + y)\(^2\)
=> ĐIỀU PHẢI CHỨNG MINH
Dấu bằng xảy ra <=> x + y = \(2\sqrt{xy}\)
<=> x = y = \(\dfrac{1}{4}\)
a)Áp dụng BĐT AM-GM ta có:
\(\left(\sqrt{x}+\sqrt{y}\right)^2=x+y+2\sqrt{xy}\)
\(\ge2\sqrt{\left(x+y\right)\cdot2\sqrt{xy}}=VP\)
Xảy ra khi \(x=y\)
b)\(BDT\Leftrightarrow x+y+z+t\ge4\sqrt[4]{xyzt}\)
Đúng với AM-GM 4 số
Xảy ra khi \(x=y=z=t\)
a) \(A=\sqrt{2+\sqrt{3}}.\sqrt{2+\sqrt{2+\sqrt{3}}}.\sqrt{2-\sqrt{2+\sqrt{3}}}\)
\(A=\sqrt{\left(2+\sqrt{3}\right)\left(\sqrt{2+\sqrt{3}}+2\right)\left(-\sqrt{2+\sqrt{3}}+2\right)}\)
\(A=\sqrt{1}\)
\(A=1\)
b)\(B=\left(\frac{\sqrt{x}}{\sqrt{xy}-y}-\frac{\sqrt{y}}{\sqrt{xy}-x}\right).\left(x\sqrt{y}-y\sqrt{x}\right)\)
\(B=\frac{\sqrt{xy}}{\sqrt{xy}-y}x\sqrt{y}+\frac{\sqrt{x}}{\sqrt{xy}-y}y\sqrt{x}+\left(-\frac{\sqrt{y}}{\sqrt{xy}-x}\right)^2x\sqrt{y}+y\sqrt{x}\)
\(B=x\frac{\sqrt{x}}{\sqrt{xy}-y}\sqrt{y}+y\frac{\sqrt{x}}{\sqrt{xy}-y}\sqrt{x}+x\frac{\sqrt{x}}{\sqrt{xy}-x}\sqrt{y}-y\sqrt{x}\frac{\sqrt{y}}{\sqrt{xy}-y}\)
\(B=\frac{-x^{\frac{5}{2}}\sqrt{y}+\sqrt{x}.y^{\frac{5}{2}}}{\left(\sqrt{xy}-y\right)\left(\sqrt{xy}-x\right)}\)
\(B=\frac{\left(\sqrt{x}.y^{\frac{5}{2}}-x^{\frac{5}{2}}\sqrt{y}\right)\left(y+\sqrt{xy}\right)\left(x+\sqrt{xy}\right)}{\left(-y^2+xy\right)\left(-x^2+xy\right)}\)
c) \(C=\sqrt{\left(3-\sqrt{5}\right)^2+\sqrt{6}-2\sqrt{5}}\)
\(C=14-6\sqrt{5}+\sqrt{6}-2\sqrt{5}\)
\(C=14-8\sqrt{5}+\sqrt{6}\)
\(C=\sqrt{14-8\sqrt{5}+\sqrt{6}}\)
a: \(B=\dfrac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\dfrac{x-1-x+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}{3}=\dfrac{\sqrt{x}-2}{3\sqrt{x}}\)
b: Để |B|=B thì B>=0
=>\(\sqrt{x}-2>=0\)
hay x>4
a)
Đặt
\(\sqrt{1+x}=a; \sqrt{1-x}=b\Rightarrow \left\{\begin{matrix} ab=\sqrt{(1+x)(1-x)}=\sqrt{1-x^2}\\ a\geq b\\ a^2+b^2=2\end{matrix}\right.\)
Khi đó:
\(A=\frac{\sqrt{1-\sqrt{1-x^2}}(\sqrt{(1+x)^3}+\sqrt{(1-x)^3})}{2-\sqrt{1-x^2}}\)
\(=\frac{\sqrt{\frac{a^2+b^2}{2}-ab}(a^3+b^3)}{a^2+b^2-ab}=\frac{\sqrt{\frac{a^2+b^2-2ab}{2}}(a+b)(a^2-ab+b^2)}{a^2+b^2-ab}\)
\(=\sqrt{\frac{a^2-2ab+b^2}{2}}(a+b)=\sqrt{\frac{(a-b)^2}{2}}(a+b)=\frac{1}{\sqrt{2}}|a-b|(a+b)\)
\(=\frac{1}{\sqrt{2}}(a-b)(a+b)=\frac{1}{\sqrt{2}}(a^2-b^2)=\frac{1}{\sqrt{2}}[(1+x)-(1-x)]=\sqrt{2}x\)
Sửa đề: \(\frac{25}{(x+z)^2}=\frac{16}{(z-y)(2x+y+z)}\)
Ta có:
Áp dụng tính chất dãy tỉ số bằng nhau thì:
\(k=\frac{a}{x+y}=\frac{5}{x+z}=\frac{a+5}{2x+y+z}=\frac{5-a}{z-y}\) ($k$ là một số biểu thị giá trị chung)
Khi đó:
\(\frac{16}{(z-y)(2x+y+z)}=\frac{25}{(x+z)^2}=(\frac{5}{x+z})^2=k^2\)
Mà: \(k^2=\frac{a+5}{2x+y+z}.\frac{5-a}{z-y}=\frac{25-a^2}{(2x+y+z)(z-y)}\)
Do đó: \(\frac{16}{(z-y)(2x+y+z)}=\frac{25-a^2}{(2x+y+z)(z-y)}\Rightarrow 16=25-a^2\)
\(\Rightarrow a^2=9\Rightarrow a=\pm 3\)
Suy ra:
\(Q=\frac{a^6-2a^5+a-2}{a^5+1}=\frac{a^5(a-2)+(a-2)}{a^5+1}=\frac{(a-2)(a^5+1)}{a^5+1}=a-2=\left[\begin{matrix}
1\\
-5\end{matrix}\right.\)
Lời giải:
Từ \(xy+x+y=1\Rightarrow \left\{\begin{matrix} x^2+1=x^2+xy+x+y=x(x+y)+(x+y)=(x+1)(x+y)\\ y^2+1=y^2+xy+x+y=y(x+y)+(x+y)=(y+1)(x+y)\end{matrix}\right.\)
Mà \(xy+x+y=1\Rightarrow x(y+1)+(y+1)=2\Rightarrow (x+1)(y+1)=2\)
Do đó:
\(x\sqrt{\frac{2(y^2+1)}{x^2+1}}+y\sqrt{\frac{2(x^2+1)}{y^2+1}}+\sqrt{\frac{(x^2+1)(y^2+1)}{2}}\)
\(=x\sqrt{\frac{(x+1)(y+1)(y+1)(x+y)}{(x+1)(x+y)}}+y\sqrt{\frac{(x+1)(y+1)(x+1)(x+y)}{(y+1)(x+y)}}+\sqrt{\frac{(x+1)(x+y)(y+1)(x+y)}{(x+1)(y+1)}}\)
\(=x\sqrt{(y+1)^2}+y\sqrt{(x+1)^2}+\sqrt{(x+y)^2}\)
\(=x(y+1)+y(x+1)+x+y=2xy+2x+2y=2(xy+x+y)=2.1=2\)