Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 320

\[1)\ y = x^{4} - 4x^{3} - 8x^{2} + 1;\]

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

\[= 4x^{3} - 12x^{2} - 16x.\]

\[4x^{3} - 12x^{2} - 16x = 0\]

\[4x\left( x^{2} - 3x - 4 \right) = 0\]

\[D = 9 + 16 = 25\]

\[x_{1} = \frac{3 - 5}{2} = - 1;\text{\ \ }\]

\[x_{2} = \frac{3 + 5}{2} = 4.\]

\[Ответ:\ - 1;\ 0;\ 4.\]

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

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

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

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

\[4x\left( 4x^{2} - 1 \right) = 0\]

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

\[x_{1} = - 0,5;\ x_{2} = 0;\ x_{3} = 0,5.\]

\[Ответ:\ - 0,5;\ 0;\ 0,5.\]

\[3)\ y = \frac{x}{3} + \frac{12}{x};\]

\[y^{'} = \frac{1}{3} + 12 \bullet \left( - \frac{1}{x^{2}} \right) = \frac{x^{2} - 36}{3x^{2}}.\]

\[x^{2} - 36 = 0\]

\[x^{2} = 36\]

\[x = \pm 6.\]

\[Ответ:\ - 6;\ 6.\]

\[4)\ y = \cos{2x} + 2\cos x;\]

\[y^{'} = - 2\sin{2x} - 2\sin x.\]

\[- 2\sin{2x} - 2\sin x = 0\]

\[- 4\sin x \bullet \cos x - 2\sin x = 0\]

\[- 2\sin x \bullet \left( 2\cos x + 1 \right) = 0\]

\[\sin x = 0\]

\[x = \pi n.\]

\[\cos x = - \frac{1}{2}\]

\[x = \pm \frac{2\pi}{3} + 2\pi n.\]

\[Ответ:\ \ \pi n;\ \pm \frac{2\pi}{3} + 2\pi n.\]