\(\cos \dfrac{{\pi \left( {4x-7} \right)}}{3} = \dfrac{1}{2}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\dfrac{{\pi \left( {4x-7} \right)}}{3} = \pm \dfrac{\pi }{3} + 2\pi k\,\left| {\, \cdot 3\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\pi \left( {4x-7} \right) = \pm \pi + 6\pi k\,\left| {:\pi \,\,\,\,\, \Leftrightarrow } \right.} \right.\)
\( \Leftrightarrow \,\,\,\,\,4x-7 = \pm 1 + 6k\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{4x = 8 + 6k,}\\{4x = 6 + 6k}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{x = 2 + 1,5k,\,\,\,\,}\\{x = 1,5 + 1,5k,}\end{array}} \right.\,\,\,\,\,k \in Z.\)
Рассмотрим \(x = 2 + 1,5k,\,\,\,k\, \in \,Z\). Если \(k = -1\), то \(x = 0,5\); если \(k = -2\), то \(x = -1.\)
Рассмотрим \(x = 1,5 + 1,5k,\,\,\,k\, \in \,Z\). Если \(k = 0\), то \(x = 1,5\); если \(k = -1\), то \(x = 0.\)
Следовательно, наименьший положительный корень \(x = 0,5.\)
Ответ: 0,5.