$\begin{array}{l} A = \sqrt {9{x^2} - 6x + 1} + \sqrt {9{x^2} + 12x + 4} \\ = \sqrt {{{\left( {3x - 1} \right)}^2}} + \sqrt {{{\left( {3x + 2} \right)}^2}} \\ = \left| {3x - 1} \right| + \left| {3x + 2} \right| = \left| {3x + 2} \right| + \left| {1 - 3x} \right| \ge \left| {3x + 2 + 1 - 3x} \right| = 3\\ \Rightarrow \min y = 3\\ ' = ' \Leftrightarrow \left( {3x + 2} \right)\left( {1 - 3x} \right) \ge 0\\ \Leftrightarrow - \dfrac{2}{3} \le x \le \dfrac{1}{3} \end{array}$