Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 1557

\[\boxed{\mathbf{1557}\mathbf{.}}\]

\[f(x) = x^{3} + x^{2} + x\sqrt{3};\]

\[g(x) = x\sqrt{3} + 1;\]

\[f^{'}(x) = \left( x^{3} \right)^{'} + \left( x^{2} \right)^{'} + \sqrt{3}(x)^{'} =\]

\[= 3x^{2} + 2x + \sqrt{3};\]

\[g^{'}(x) = \sqrt{3}(x)^{'} + (1)^{'} =\]

\[= \sqrt{3} + 0 = \sqrt{3}.\]

\[При\ f^{'}(x) \leq g^{'}(x):\]

\[3x^{2} + 2x + \sqrt{3} \leq \sqrt{3}\]

\[3x^{2} + 2x \leq 0\]

\[(3x + 2) \bullet x \leq 0\]

\[- \frac{2}{3} \leq x \leq 0.\]

\[Ответ:\ \ x \in \left\lbrack - \frac{2}{3};\ 0 \right\rbrack.\]