\(F=\dfrac{1}{\sqrt{2}}\sqrt{\left(3x-2\right)-2\sqrt{x}\sqrt{3x-2}+x}+\dfrac{1}{\sqrt{2}}\sqrt{9x+2.3\sqrt{x}\sqrt{3x-2}+3x-2}\)
\(=\dfrac{1}{\sqrt{2}}\sqrt{\left(\sqrt{3x-2}-\sqrt{x}\right)^2}+\dfrac{1}{\sqrt{2}}\sqrt{\left(3\sqrt{x}+\sqrt{3x-2}\right)^2}\)
\(=\dfrac{1}{\sqrt{2}}\left(\left|\sqrt{3x-2}-\sqrt{x}\right|+3\sqrt{x}+\sqrt{3x-2}\right)\)
Do \(\sqrt{3x-2}< \sqrt{x}\forall\dfrac{2}{3}< x< 1\)
\(F=\dfrac{1}{\sqrt{2}}\left(\sqrt{x}+3\sqrt{x}\right)=2\sqrt{2x}\)
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