a) \(P=\left(\frac{5\sqrt{x}}{x-4}-\frac{\sqrt{x}}{\sqrt{x}-2}+\frac{\sqrt{x}}{\sqrt{x}+2}\right):\left(2-\sqrt{x}\right)\)
\(P=\frac{5\sqrt{x}-\sqrt{x}\left(\sqrt{x}+2\right)+\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\cdot\frac{1}{2-\sqrt{x}}\)
\(P=\frac{5\sqrt{x}-x-2\sqrt{x}+x-2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\cdot\frac{1}{2-\sqrt{x}}\)
\(P=\frac{-\sqrt{x}}{\left(\sqrt{x}-2\right)^2\left(\sqrt{x}+2\right)}\)
\(P=\frac{-x-2\sqrt{x}}{\left(x-4\right)^2}\)
b) \(P=\frac{1}{3}\Leftrightarrow\frac{-x-2\sqrt{x}}{\left(x-4\right)^2}=\frac{1}{3}\)
\(\Leftrightarrow-3x-6\sqrt{x}=x^2-8x+16\)
\(\Leftrightarrow x^2-5x+6\sqrt{x}+16=0\)
\(\Leftrightarrow x^2-5x+\frac{25}{4}+6\sqrt{x}+\frac{39}{4}=0\)
\(\Leftrightarrow\left(x-\frac{5}{2}\right)^2+6\sqrt{x}+\frac{39}{4}=0\) ( vô lý )
Vậy pt vô nghiệm.
c) \(P=\frac{-x-2\sqrt{x}}{\left(x-4\right)^2}=\frac{-\sqrt{x}\left(\sqrt{x}+2\right)^2}{\left(x-4\right)^2}\le0\forall x\)
Dấu "=" xảy ra \(\Leftrightarrow x=0\)
Vậy GTLN của P là 0 khi x = 0.
