Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 868

\[\boxed{\mathbf{868}\mathbf{.}}\]

\[f(x) = x^{3} - x - 1\]

\[g(x) = 3x^{2} - 4x + 1:\]

\[f^{'}(x) = \left( x^{3} \right)^{'} - (x + 1)^{'} =\]

\[= 3x^{2} - 1\]

\[g^{'}(x) = 3 \bullet \left( x^{2} \right)^{'} - (4x - 1)^{'} =\]

\[= 3 \bullet 2x - 4 = 6x - 4.\]

\[3x^{2} - 1 = 6x - 4\]

\[3x^{2} - 6x + 3 = 0\]

\[x^{2} - 2x + 1 = 0\]

\[(x - 1)^{2} = 0\]

\[x - 1 = 0\]

\[x = 1.\]

\[f(1) = 1^{3} - 1 - 1 = - 1\]

\[g(1) = 3 \bullet 1^{2} - 4 \bullet 1 + 1 = 0.\]

\[Первая\ касательная:\]

\[f^{'}(1) = 3 \bullet 1^{2} - 1 = 3 - 1 = 2\]

\[y = - 1 + 2(x - 1) =\]

\[= - 1 + 2x - 2 = 2x - 3.\]

\[Вторая\ касательная:\]

\[g^{'}(1) = 6 \bullet 1 - 4 = 6 - 4 = 2\]

\[y = 0 + 2(x - 1) = 2x - 2.\]

\[Ответ:\ \ (1\ - 1),\ \ \ y = 2x - 3\ \ \]

\[\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(1\ 0),\ \ \ y = 2x - 2.\]