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ta có: \(\left(xy+\sqrt{\left(1+x^2\right)\left(1+y^2\right)}\right)^2=1^2\)
\(\Leftrightarrow2x^2y^2+x^2+y^2+2xy\sqrt{\left(1+x^2\right)\left(1+y^2\right)}=0\)
\(\Leftrightarrow\left(x\sqrt{1+y^2}+y\sqrt{1+x^2}\right)^2=0\)
\(\Leftrightarrow x\sqrt{1+y^2}+y\sqrt{1+x^2}=0\left(đpcm\right)\)
a, \(\frac{2+\sqrt{3}}{2+\sqrt{4+2\sqrt{3}}}+\frac{2-\sqrt{3}}{2-\sqrt{4-2\sqrt{3}}}\)
\(=\frac{2+\sqrt{3}}{2+\sqrt{\left(\sqrt{3}+1\right)^2}}+\frac{2-\sqrt{3}}{2-\sqrt{\left(\sqrt{3}-1\right)^2}}\)
\(=\frac{2+\sqrt{3}}{2+\sqrt{3}+1}+\frac{2-\sqrt{3}}{2-\sqrt{3}+1}\)
\(=\frac{2+\sqrt{3}}{3+\sqrt{3}}+\frac{2-\sqrt{3}}{3-\sqrt{3}}\)
\(=\frac{\left(2+\sqrt{3}\right)\left(3-\sqrt{3}\right)+\left(2-\sqrt{3}\right)\left(3+\sqrt{3}\right)}{\left(3+\sqrt{3}\right)\left(3-\sqrt{3}\right)}\)
\(=\frac{6+\sqrt{3}-3+6-\sqrt{3}-3}{9-3}=\frac{6}{6}=1\)
b, \(\frac{1}{x+\sqrt{x}}+\frac{2\sqrt{x}}{x-1}-\frac{1}{x-\sqrt{x}}\)
\(=\frac{1}{\sqrt{x}\left(\sqrt{x}+1\right)}+\frac{2\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x-1}\right)}-\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\)
\(=\frac{\sqrt{x}-1+2x-\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{2\left(x-1\right)}{\sqrt{x}\left(x-1\right)}=\frac{2}{\sqrt{x}}\)
Chứng minh rằng nếu x ≥ 2 thì: \(\sqrt{x-1+2\sqrt{x-2}}+\sqrt{x-1-2\sqrt{x-2}}\) ≥ 2
Mong mn giúp đỡ.
VT=|căn(x-2)+1|+|căn (x-2)-1|
=|căn (x-2)+1|+|1-căn x-2|>=|căn(x-2)+1+1-căn(x-2)|=2
Answer:
a. \(P=\left(\frac{\sqrt{x}-2}{x-1}-\frac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right)\left(\frac{1-x}{\sqrt{2}}\right)^2\) ĐK: \(x\ge0;x\ne1\)
\(=\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)-\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}.\frac{\left(1-x\right)^2}{2}\)
\(=\frac{-2\sqrt{x}}{\sqrt{x}+1}.\frac{x-1}{2}\)
\(=\frac{\sqrt{x}\left(1-x\right)}{\sqrt{x}+1}\)
\(=\frac{\sqrt{x}\left(1-\sqrt{x}\right)\left(\sqrt{x}+1\right)}{\sqrt{x}+1}\)
\(=\sqrt{x}\left(1-\sqrt{x}\right)\)
b. Vì \(0< x< 1\Rightarrow\hept{\begin{cases}\sqrt{x}\ge0\\1-\sqrt{x}>0\end{cases}}\Rightarrow\sqrt{x}\left(1-\sqrt{x}\right)>0\)
Do vậy \(\sqrt{x}\left(1-\sqrt{x}\right)>0\)
c. \(P=\sqrt{x}\left(1-\sqrt{x}\right)\)
\(=-\left(\sqrt{x}\right)^2+\sqrt{x}\)
\(=-\left(x-2\sqrt{x}.\frac{1}{2}+\frac{1}{4}\right)+\frac{1}{4}\)
\(=-\left(\sqrt{x}-\frac{1}{2}\right)^2+\frac{1}{4}\le\frac{1}{4}\forall x\)
Dấu "=" xảy ra khi \(\sqrt{x}-\frac{1}{2}=0\Rightarrow x=\frac{1}{4}\)
ta có \(\sqrt{x-1+2\sqrt{x-2}}+\sqrt{x-1-2\sqrt{x-2}}\)
\(=\sqrt{\left(\sqrt{x-2}+1\right)^2}+\sqrt{\left(\sqrt{x-2}-1\right)^2}\)
\(=\left|\sqrt{x-2}+1\right|+\left|\sqrt{x-2}-1\right|\)
Vì \(x\ge2\Rightarrow\sqrt{x-2}\ge0\Rightarrow\left\{{}\begin{matrix}\sqrt{x-2}+1\ge1\\\sqrt{x-2}-1\ge-1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left|\sqrt{x-2}+1\right|\ge1\\\left|\sqrt{x-2}-1\right|\ge1\end{matrix}\right.\)
\(\Leftrightarrow\left|\sqrt{x-2}+1\right|+\left|\sqrt{x-2}-1\right|\ge2\)
Hay A\(\ge2\) Dấu = xảy ra khi x=2
=> đpcm
ĐK: \(-1\le x\le1\)\(;\)\(x\ne0\)
\(\frac{\sqrt{1+x}+\sqrt{1-x}}{\sqrt{1+x}-\sqrt{1-x}}=\sqrt{2}\)
\(\Leftrightarrow\)\(\frac{\left(\sqrt{1+x}+\sqrt{1-x}\right)^2}{\left(\sqrt{1+x}-\sqrt{1-x}\right)\left(\sqrt{1+x}+\sqrt{1-x}\right)}=\sqrt{2}\)
\(\Leftrightarrow\)\(\frac{1+x+1-x+2\sqrt{\left(1+x\right)\left(1-x\right)}}{1+x-1+x}=\sqrt{2}\)
\(\Leftrightarrow\)\(\sqrt{1-x^2}=\sqrt{2}x-1\)
\(\Leftrightarrow\)\(1-x^2=2x^2-2\sqrt{2}x+1\)
\(\Leftrightarrow\)\(x^2-\frac{2\sqrt{2}}{3}x=0\)
\(\Leftrightarrow\)\(x\left(x-\frac{2\sqrt{2}}{3}\right)=0\)
\(\Leftrightarrow\)\(\orbr{\begin{cases}x=0\left(l\right)\\x=\frac{2\sqrt{2}}{3}\left(tm\right)\end{cases}}\)
\(\frac{x-1}{x+1}=\frac{\frac{2\sqrt{2}}{3}-1}{\frac{2\sqrt{2}}{3}+1}=\frac{\frac{2\sqrt{2}-3}{3}}{\frac{2\sqrt{2}+3}{3}}=\frac{2\sqrt{2}-3}{2\sqrt{2}+3}=12\sqrt{2}-17\) ( giống như tìm x ở trên )
@Doraemon2611.
:))