\({\rm{ctg}}\dfrac{{\pi \left( {2x-1} \right)}}{3} = \sqrt 3 \,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\dfrac{{\pi \left( {2x-1} \right)}}{3} = \dfrac{\pi }{6} + \pi k\left| { \cdot 3} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\pi \left( {2x-1} \right) = \dfrac{\pi }{2} + 3\pi k\left| {:\pi } \right.\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,2x-1 = \dfrac{1}{2} + 3k\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,2x = \dfrac{3}{2} + 3k\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,x = 0,75 + 1,5k,\,\,\,\,\,k \in Z.\)
Рассмотрим \(x = 0,75 + 1,5k,\,\,\,\,k\, \in \,Z\). Если \(k = 0\), то \(x = 0,75\); если \(k = -1\), то \(x = -0,75.\)
Следовательно, наибольший отрицательный корень \(x = -0,75.\)
Ответ: \(-0,75.\)