Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 962

\[\boxed{\mathbf{962}\mathbf{.}}\]

\[1)\ f(x) = x^{3} - 6x^{2} + 9;\ \ \ \]

\[\lbrack - 2;\ 2\rbrack;\]

\[f^{'}(x) = \left( x^{3} \right)^{'} - 6 \bullet \left( x^{2} \right)^{'} + (9)^{'};\]

\[f^{'}(x) = 3x^{2} - 6 \bullet 2x + 0 =\]

\[= 3x^{2} - 12x.\]

\[Точки\ экстремума:\]

\[3x^{2} - 12x = 0\]

\[3x \bullet (x - 4) = 0\]

\[x_{1} = 0\ и\ x_{2} = 4.\]

\[f( - 2) = ( - 2)^{3} - 6 \bullet ( - 2)^{2} + 9 =\]

\[= - 8 - 24 + 9 = - 23;\]

\[f(0) = 0^{3} - 6 \bullet 0^{2} + 9 = 9;\]

\[f(2) = 2^{3} - 6 \bullet 2^{2} + 9 =\]

\[= 8 - 24 + 9 = - 7.\]

\[Ответ:\ \ y_{\min} = - 23;\ \ y_{\max} = 9.\]

\[2)\ f(x) = x^{3} + 6x^{2} + 9x;\ \ \]

\[\lbrack - 4;\ 0\rbrack\]

\[f^{'}(x) = \left( x^{3} \right)^{'} + 6 \bullet \left( x^{2} \right)^{'} + (9x)^{'};\]

\[f^{'}(x) = 3x^{2} + 6 \bullet 2x + 9 =\]

\[= 3x^{2} + 12x + 9.\]

\[Точки\ экстремума:\]

\[3x^{2} + 12x + 9 = 0\]

\[x^{2} + 4x + 3 = 0\]

\[D = 4^{2} - 4 \bullet 3 = 16 - 12 = 4\]

\[x_{1} = \frac{- 4 - 2}{2} = - 3\ \ и\ \ \]

\[x_{2} = \frac{- 4 + 2}{2} = - 1.\]

\[f( - 4) =\]

\[= ( - 4)^{3} + 6 \bullet ( - 4)^{2} + 9 \bullet ( - 4) =\]

\[= - 64 + 96 - 36 = - 4;\]

\[f( - 3) =\]

\[= ( - 3)^{3} + 6 \bullet ( - 3)^{2} + 9 \bullet ( - 3) =\]

\[= - 27 + 54 - 27 = 0;\]

\[f( - 1) =\]

\[= ( - 1)^{3} + 6 \bullet ( - 1)^{2} + 9 \bullet ( - 1) =\]

\[= - 1 + 6 - 9 = - 4.\]

\[f(0) = 0^{3} + 6 \bullet 0^{2} + 9 \bullet 0 = 0;\]

\[Ответ:\ \ y_{\min} = - 4;\ \ y_{\max} = 0.\]

\[3)\ f(x) = x^{4} - 2x^{2} + 3;\ \ \ \ \]

\[\lbrack - 4;\ 3\rbrack\]

\[f^{'}(x) = \left( x^{4} \right)^{'} - 2 \bullet \left( x^{2} \right)^{'} + (3)^{'};\]

\[f^{'}(x) = 4x^{3} - 2 \bullet 2x + 0 =\]

\[= 4x^{3} - 4x.\]

\[Точки\ экстремума:\]

\[4x^{3} - 4x = 0\]

\[4x \bullet \left( x^{3} - 1 \right) = 0\]

\[x_{1} = 0\ и\ x_{2} = 1.\]

\[f( - 4) =\]

\[= ( - 4)^{4} - 2 \bullet ( - 4)^{2} + 3 =\]

\[= 256 - 32 + 3 = 227;\]

\[f(0) = 0^{4} - 2 \bullet 0^{2} + 3 = 3;\]

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

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

\[f(3) = 3^{4} - 2 \bullet 3^{2} + 3 =\]

\[= 81 - 18 + 3 = 66.\]

\[Ответ:\ \ y_{\min} = 2;\ \ y_{\max} = 227.\]

\[4)\ f(x) = x^{4} - 8x^{2} + 5;\ \ \ \lbrack - 3;\ 2\rbrack\]

\[f^{'}(x) = \left( x^{4} \right)^{'} - 8 \bullet \left( x^{2} \right)^{'} + (5)^{'};\]

\[f^{'}(x) = 4x^{3} - 8 \bullet 2x + 0 =\]

\[= 4x^{3} - 16x.\]

\[Точки\ экстремума:\]

\[4x^{3} - 16x = 0\]

\[4x \bullet \left( x^{2} - 4 \right) = 0\]

\[x_{1} = \pm 2\ и\ x_{2} = 0.\]

\[f( - 3) =\]

\[= ( - 3)^{4} - 8 \bullet ( - 3)^{2} + 5 =\]

\[= 81 - 72 + 5 = 14;\]

\[f( \pm 2) =\]

\[= ( \pm 2)^{4} - 8 \bullet ( \pm 2)^{2} + 5 =\]

\[= 16 - 32 + 5 = - 11;\]

\[f(0) = 0^{4} - 8 \bullet 0^{2} + 5 = 5.\]

\[Ответ:\ \ y_{\min} = - 11;\ \ y_{\max} = 14.\]