\(g\left( x \right) = \sqrt[3]{x}\left( {x - 1} \right) = {x^{\frac{1}{3}}}\left( {x - 1} \right) = {x^{\frac{4}{3}}} - {x^{\frac{1}{3}}}\)\(g"\left( x \right) = \dfrac{4}{3}{x^{\frac{1}{3}}} - \dfrac{1}{3}{x^{ - \;\frac{2}{3}}}\)\(g""\left( x \right) = \dfrac{4}{9}{x^{ - \;\frac{2}{3}}} + \dfrac{2}{9}{x^{ - \;\frac{5}{3}}} = \dfrac{2}{9}{x^{ - \;\frac{5}{3}}}\left( {2x + 1} \right) = \dfrac{{2\left( {2x + 1} \right)}}{{9{x^{\frac{5}{3}}}}} = \dfrac{{2\left( {2x + 1} \right)}}{{9\sqrt[3]{{{x^5}}}}}\)\(g""\left( x \right) = \dfrac{{2\left( {2x + 1} \right)}}{{9\sqrt[3]{{{x^5}}}}} = 0\;\;\;\; \Rightarrow \;\;\;\;2x + 1 = 0\;\;\;\; \Rightarrow \;\;\;\;x = - \dfrac{1}{2}\)Check the sign (+ or -) of \(g""\left( x \right)\) on either side of \(x = - \dfrac{1}{2}\)\(g""\left( { - 1} \right) = \dfrac{{2\left( { - 2 + 1} \right)}}{{9\sqrt[3]{{ - 1}}}} = \dfrac{{ - 2}}{{ - 9}} = \dfrac{2}{9} > 0\) graph of \(g\) is concave up at \(x = - 1\)\(g""\left( { - \dfrac{1}{8}} \right) = \dfrac{{2\left( { - \dfrac{1}{4} + 1} \right)}}{{9\sqrt[3]{{ - \dfrac{1}{8}}}}} = \dfrac{{\dfrac{3}{2}}}{{ - \dfrac{9}{2}}} = - \dfrac{1}{3} < 0\) graph of \(g\) is concave down at \(x = - \dfrac{1}{8}\)Hence, since the concavity of the graph of \(g\) changes from up to down at \(x = - \dfrac{1}{2}\) then there is a point of inflexion at \(x = - \dfrac{1}{2}\).\(\displaystyle\ g\left( { - \frac{1}{2}} \right) = \sqrt[3]{{ - \frac{1}{2}}}\left( { - \frac{1}{2} - 1} \right) = - \frac{1}{{\sqrt[3]{2}}}\left( { - \frac{3}{2}} \right) = \frac{3}{{2\sqrt[3]{2}}} \cdot \frac{{\sqrt[3]{4}}}{{\sqrt[3]{4}}} = \frac{{3\sqrt[3]{4}}}{{2\sqrt[3]{8}}} = \frac{{3\sqrt[3]{4}}}{4}\)■ The exact coordinates of the first point of inflexion are \(\displaystyle\ \left( { - \frac{1}{2},\frac{{3\sqrt[3]{4}}}{4}} \right)\).