a) \(A=\left(\dfrac{\sqrt{x}-2}{x-1}-\dfrac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right).\dfrac{\left(1-x\right)^2}{2}\)
\(A=\left(\dfrac{\sqrt{x}-2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}-\dfrac{\sqrt{x}+2}{\left(\sqrt{x}+1\right)^2}\right).\dfrac{\left(1-x\right)^2}{2}\)
\(A=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)-\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{-\left(1-x\right)\left(\sqrt{x}+1\right)}.\dfrac{\left(1-x\right)^2}{2}\)
\(A=\dfrac{x-\sqrt{x}-2-x-\sqrt{x}+2}{-\left(\sqrt{x}+1\right)}.\dfrac{1-x}{2}\)
\(A=\dfrac{\sqrt{x}\left(1-\sqrt{x}\right)\left(1+\sqrt{x}\right)}{\left(\sqrt{x}+1\right)}=\sqrt{x}-x\)
b) Để A dương
\(\sqrt{x}-x>0\)
\(\sqrt{x}\left(1-\sqrt{x}\right)>0\)
\(\Rightarrow0< x< 1\)
c) \(A=-\left(x-\sqrt{x}+\dfrac{1}{4}\right)+\dfrac{1}{4}\)
\(A=-\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{1}{4}\)
\(A_{max}=\dfrac{1}{4}\Leftrightarrow\sqrt{x}=\dfrac{1}{2}\Leftrightarrow x=\dfrac{1}{4}\)
\(a,A=\left(\dfrac{\sqrt{x}-2}{x-1}-\dfrac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right)\cdot\dfrac{\left(1-x\right)^2}{2}\left(x\ge0,x\ne1\right)\\ A=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)-\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}\cdot\dfrac{\left(x-1\right)^2}{2}\\ A=\dfrac{x-\sqrt{x}-1-x-\sqrt{x}+1}{\left(x-1\right)\left(\sqrt{x}+1\right)}\cdot\dfrac{\left(x-1\right)^2}{2}\\ A=\dfrac{-2\sqrt{x}\left(x-1\right)}{2\left(\sqrt{x}+1\right)}\\ A=\dfrac{-2\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{2\left(\sqrt{x}+1\right)}=-\sqrt{x}\left(\sqrt{x}-1\right)\)
\(b,\) Để A dương \(\Leftrightarrow A>0\Leftrightarrow-\sqrt{x}\left(\sqrt{x}-1\right)>0\)
\(\Leftrightarrow\sqrt{x}-1< 0\left(-\sqrt{x}< 0\Leftrightarrow0< x\right)\\ \Leftrightarrow0< x< 1\)
\(c,A=-\sqrt{x}\left(\sqrt{x}-1\right)\\ =\sqrt{x}-x\\ =-\left(x-\sqrt{x}+\dfrac{1}{4}\right)+\dfrac{1}{4}\\ =-\left(\sqrt{x}-\dfrac{1}{2}\right)^2+\dfrac{1}{4}\le\dfrac{1}{4}\)
Dấu \("="\Leftrightarrow\sqrt{x}=\dfrac{1}{2}\Leftrightarrow x=\dfrac{1}{4}\)
a: Ta có: \(A=\left(\dfrac{\sqrt{x}-2}{x-1}-\dfrac{\sqrt{x}+2}{x+2\sqrt{x}+1}\right)\cdot\dfrac{\left(x-1\right)^2}{2}\)
\(=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)-\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\cdot\left(\sqrt{x}+1\right)^2}\cdot\dfrac{\left(\sqrt{x}-1\right)^2\cdot\left(\sqrt{x}+1\right)^2}{2}\)
\(=\dfrac{x-\sqrt{x}-2-x-\sqrt{x}-2}{1}\cdot\dfrac{\sqrt{x}-1}{2}\)
\(=-\sqrt{x}\left(\sqrt{x}-1\right)\)
