Решебник по алгебре и начала математического анализа 10 класс Алимов Задание 1591

\[\boxed{\mathbf{1591}\mathbf{.}}\]

\[f(x) = x^{3} + ax^{2} + bx + c\]

\[Координаты\ точек:\]

\[A\left( - 2 - l;\ f( - 2 - l) \right);\]

\[B\left( l - 2;\ f(l - 2) \right).\]

\[f^{'}(x) =\]

\[= \left( x^{3} \right)^{'} + a\left( x^{2} \right)^{'} + (bx + c)^{'} =\]

\[= 3x^{2} + 2ax + b.\]

\[Касательные\ параллельны:\]

\[3( - 2 - l)^{2} + 2a( - 2 - l) + b =\]

\[= 3(l - 2)^{2} + 2a(l - 2) + b\]

\[3(l + 2)^{2} - 4a - 2al =\]

\[= 3(l - 2)^{2} + 2al - 4a\]

\[3\left( l^{2} + 4l + 4 \right) - 2al =\]

\[= 3\left( l^{2} - 4l + 4 \right) + 2al\]

\[3l^{2} + 12l + 12 - 2al =\]

\[= 3l^{2} - 12l + 12 + 2al\]

\[12l + 12l - 2al - 2al = 0\]

\[24l - 4al = 0\]

\[6l - al = 0\]

\[l(6 - a) = 0\]

\[a = 6;\]

\[\left( \text{l\ }не\ может\ быть\ равным\ нулю \right).\]

\[Уравнение\ в\ точке\ t:\]

\[f^{'}(t) = 3t^{2} + 2 \bullet 6t + b =\]

\[= 3t^{2} + 12t + b;\]

\[f(t) = t^{3} + 6t^{2} + bt + c;\]

\[y = b \bullet 0 + c - 2t^{3} + 3t^{2} \bullet 0 - 6t^{2} + 12t \bullet 0 =\]

\[= c - 2t^{3} - 6t^{2}.\]

\[В\ точке\ A\ через\ (0;\ 1):\]

\[1 = c - 2( - 2 - l)^{3} - 6( - 2 - l)^{2}\]

\[c + 2l^{3} + 6l^{2} - 9 = 0\]

\[c = - 2l^{3} - 6l^{2} + 9.\]

\[В\ точке\ B\ через\ (0;\ 5):\]

\[5 = c - 2(l - 2)^{3} - 6(l - 2)^{2}\]

\[c - 2l^{3} + 6l^{2} - 13 = 0\]

\[c = 2l^{3} - 6l^{2} + 13.\]

\[Найдем\ l:\]

\[- 2l^{3} - 6l^{2} + 9 = 2l^{3} - 6l^{2} + 13\]

\[- 4l^{3} = 4\]

\[l^{3} = - 1\]

\[l = - 1.\]

\[Найдем\ c:\]

\[c = 2 \bullet ( - 1)^{3} - 6 \bullet ( - 1)^{2} + 13 =\]

\[= - 2 - 6 + 13 = 5.\]

\[Точка\ A:\]

\[- 2 - l = - 2 + 1 = - 1;\]

\[f( - 1) =\]

\[= ( - 1)^{3} + 6 \bullet ( - 1)^{2} + b \bullet ( - 1) + 5 =\]

\[= - 1 + 6 - b + 5 = 10 - b;\]

\[A( - 1;\ 10 - b);\]

\[Точка\ B:\]

\[l - 2 = - 1 - 2 = - 3;\]

\[f( - 3) =\]

\[= ( - 3)^{3} + 6 \bullet ( - 3)^{2} + b \bullet ( - 3) + 5 =\]

\[= - 27 + 54 - 3b + 5 = 32 - 3b;\]

\[B( - 3;\ 32 - 3b).\]

\[Симметричны\ прямой\ x = - 2:\]

\[10 - b = 32 - 3b\]

\[2b = 22\]

\[b = 11.\]

\[Ответ:\ \ a = 6;\ \ b = 11;\ \ c = 5.\]