\(1,x=16\Rightarrow A=\dfrac{16-1}{\sqrt{16}}=\dfrac{15}{4}\)
\(2,B=\dfrac{\sqrt{x}+1}{\sqrt{x}-1}-\dfrac{\sqrt{x}-1}{\sqrt{x}+1}+4\sqrt{x}\left(dl:x>0,x\ne1\right)\)
\(=\dfrac{\left(\sqrt{x}+1\right)^2-\left(\sqrt{x}-1\right)^2+4\sqrt{x}\left(x-1\right)}{x-1}\\ =\dfrac{x+2\sqrt{x}+1-x+2\sqrt{x}-1+4x\sqrt{x}-4\sqrt{x}}{x-1}\\ =\dfrac{4x\sqrt{x}}{x-1}\)
\(3,P=A.B=\dfrac{x-1}{\sqrt{x}}.\dfrac{4x\sqrt{x}}{x-1}=4x\)
\(\sqrt{P}>P\Leftrightarrow\sqrt{4x}>4x\Leftrightarrow\left(\sqrt{4x}\right)^2>\left(4x\right)^2\Leftrightarrow4x>16x^2\Leftrightarrow4x-16x^2>0\Leftrightarrow4x\left(1-4x\right)>0\)
\(\Leftrightarrow\left[{}\begin{matrix}4x>0\\1-4x>0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x>0\\x< \dfrac{1}{4}\end{matrix}\right.\)
Vậy \(S=\left\{x|0< x< \dfrac{1}{4}\right\}\) thì \(\sqrt{P}>P\)
\(4,\left|P\right|>P\Leftrightarrow\left|4x\right|>4x\)
\(TH_1:x\ge0\\4x>4x\Leftrightarrow4x-4x>0\Leftrightarrow0>0\left(VL\right) \)
\(TH_2:x< 0\\ -4x>4x\Leftrightarrow-4x-4x>0\Leftrightarrow-8x>0\Leftrightarrow x< 0\)
Vậy \(x< 0\) thì \(\left|P\right|>P\)
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