ГДЗ по алгебре 11 класс Никольский Параграф 5. Применение производной Задание 31

\[\boxed{\mathbf{31.}}\]

\[\textbf{а)}\ f(x) = x^{2} + 4x - 12;\]

\[\ \ f'\left( x_{0} \right) = 0\]

\[f^{'}(x) = 2x + 4;\]

\[2x + 4 = 0\]

\[2x = - 4\]

\[x = - 2.\]

\[Ответ:x = - 2.\]

\[\textbf{б)}\ f(x) = 3x^{2} - 12x + 11;\ \]

\[\ f'\left( x_{0} \right) = 0\]

\[f^{'}(x) = 6x - 12;\]

\[6x - 12 = 0\]

\[6x = 12\]

\[x = 2.\]

\[Ответ:x = 2.\]

\[\textbf{в)}\ f(x) = x^{3} - 12x^{2} + 36x - 1;\ \ \]

\[f'\left( x_{0} \right) = 0\]

\[f^{'}(x) = 3x^{2} - 24x + 36;\]

\[3x^{2} - 24x + 36 = 0\ \ \ \ |\ :3\]

\[x^{2} - 8x + 12 = 0\]

\[D_{1} = 16 - 12 = 4\]

\[x_{1} = 4 + 2 = 6;\]

\[x_{2} = 4 - 2 = 2.\]

\[Ответ:x = 6;x = 2.\]

\[\textbf{г)}\ f(x) = 2x^{3} + 6x^{2} - 7;\ \ \]

\[f'\left( x_{0} \right) = 0\]

\[f^{'}(x) = 6x^{2} + 12x;\]

\[6x^{2} + 12x = 0\]

\[6x(x + 2) = 0\]

\[x = 0;\ \ x = - 2.\]

\[Ответ:x = - 2;x = 0.\]