Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 729

\[\boxed{\mathbf{729}\mathbf{.}}\]

\[1)\ y = 1 - \sin x\]

\[\textbf{а)}\ D(x) = ( - \infty;\ + \infty);\]

\[\textbf{б)} - 1 \leq \sin x \leq 1;\]

\[- 1 \leq - \sin x \leq 1;\]

\[0 \leq 1 - \sin x \leq 2;\]

\[E(y) = \lbrack 0;\ 2\rbrack.\]

\[\textbf{в)}\ y(x + T) = y(x);\]

\[1 - \sin(x + T) = 1 - \sin x;\]

\[T = 2\pi.\]

\[\textbf{г)}\ Ни\ четная,\ ни\ нечетная:\]

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

\[\textbf{д)}\ 1 - \sin x = 0;\]

\[\sin x = 1;\]

\[x = \arcsin 1 + 2\pi n = \frac{\pi}{2} + 2\pi n.\]

\[\textbf{е)}\ Максимальные\ значения:\]

\[1 - \sin x = 2;\]

\[\sin x = 1 - 2;\]

\[\sin x = - 1;\]

\[x = \arcsin( - 1) + 2\pi n\]

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

\[\textbf{ж)}\ Минимальные\ значения:\]

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

\[\textbf{з)}\ Возрастает:\ \]

\[\frac{\pi}{2} + 2\pi n < x < \frac{3\pi}{2} + 2\pi n;\]

\[убывает:\]

\[- \frac{\pi}{2} + 2\pi n < x < \frac{\pi}{2} + 2\pi n;\]

\[положительна:\]

\[x \neq \frac{\pi}{2} + 2\pi n.\]

\[2)\ y = 2 + \sin x\]

\[\textbf{а)}\ D(x) = ( - \infty;\ + \infty);\]

\[\textbf{б)}\ - 1 \leq \sin x \leq 1;\]

\[1 \leq 2 + \sin x \leq 3;\]

\[E(y) = \lbrack 1;\ 3\rbrack.\]

\[\textbf{в)}\ y(x + T) = y(x);\]

\[2 + \sin(x + T) = 2 + \sin x;\]

\[T = 2\pi.\]

\[\textbf{г)}\ Ни\ четная,\ ни\ нечетная:\]

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

\[\textbf{д)}\ 2 + \sin x = 0\]

\[нет\ корней.\]

\[\textbf{е)}\ Максимальные\ значения:\]

\[2 + \sin x = 3;\]

\[\sin x = 1;\]

\[x = \arcsin 1 + 2\pi n = \frac{\pi}{2} + 2\pi n.\]

\[\textbf{ж)}\ Минимальные\ значения:\]

\[2 + \sin x = 1;\]

\[\sin x = - 1;\]

\[x = \arcsin( - 1) + 2\pi n\]

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

\[\textbf{з)}\ Возрастает:\]

\[- \frac{\pi}{2} + 2\pi n < x < \frac{\pi}{2} + 2\pi n;\]

\[убывает:\ \]

\[\frac{\pi}{2} + 2\pi n < x < \frac{3\pi}{2} + 2\pi n;\]

\[положительна\ при\ x \in R.\]

\[3)\ y = \sin{3x}\]

\[\textbf{а)}\ D(x) = ( - \infty;\ + \infty);\]

\[\textbf{б)}\ - 1 \leq \sin{3x} \leq 1;\]

\[E(y) = \lbrack - 1;\ 1\rbrack.\]

\[\textbf{в)}\ y(x + T) = y(x);\]

\[\sin\left( 3 \bullet (x + T) \right) = \sin{3x};\]

\[\sin(3x + 3T) = \sin{3x};\]

\[3T = 2\pi;\]

\[T = \frac{2\pi}{3}.\]

\[\textbf{г)}\ Функция\ нечетная:\]

\[y( - x) = \sin( - 3x) =\]

\[= - \sin{3x} = - y(x).\]

\[\textbf{д)}\ \sin{3x} = 0;\]

\[3x = \arcsin 0 + \pi n = \pi n;\]

\[x = \frac{\text{πn}}{3}.\]

\[\textbf{е)}\ Максимальные\ значения:\]

\[\sin{3x} = 1\]

\[3x = \arcsin 1 + 2\pi n\]

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

\[x = \frac{1}{3} \bullet \left( \frac{\pi}{2} + 2\pi n \right) = \frac{\pi}{6} + \frac{2\pi n}{3}.\]

\[\textbf{ж)}\ Минимальные\ значения:\]

\[\sin{3x} = - 1\]

\[3x = - \arcsin 1 + 2\pi n\]

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

\[x = \frac{1}{3} \bullet \left( - \frac{\pi}{2} + 2\pi n \right)\]

\[x = - \frac{\pi}{6} + \frac{2\pi n}{3}.\]

\[\textbf{з)}\ Возрастает:\]

\[- \frac{\pi}{6} + \frac{2\pi n}{3} < x < \frac{\pi}{6} + \frac{2\pi n}{3};\]

\[убывает:\]

\[\frac{\pi}{6} + \frac{2\pi n}{3} < x < \frac{\pi}{2} + \frac{2\pi n}{3};\]

\[положительна:\]

\[\frac{2\pi n}{3} < x < \frac{\pi}{3} + \frac{2\pi n}{3};\]

\[отрицательна:\]

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

\[4)\ y = 2\sin x\]

\[\textbf{а)}\ D(x) = ( - \infty;\ + \infty);\]

\[\textbf{б)}\ - 1 \leq \sin x \leq 1;\]

\[- 2 \leq 2\sin x \leq 2;\]

\[E(y) = \lbrack - 2;\ 2\rbrack.\]

\[\textbf{в)}\ y(x + T) = y(x);\]

\[2\sin(x + T) = 2\sin x;\]

\[T = 2\pi.\]

\[\textbf{г)}\ Функция\ нечетная:\]

\[y( - x) = 2\sin( - x) =\]

\[= - 2\sin x = - y(x).\]

\[\textbf{д)}\ 2\sin x = 0;\]

\[\sin x = 0;\]

\[x = \arcsin 0 + \pi n = \pi n.\]

\[\textbf{е)}\ Максимальные\ значения:\]

\[2\sin x = 2\]

\[\sin x = 1\]

\[x = \arcsin 1 + 2\pi n = \frac{\pi}{2} + 2\pi n.\]

\[\textbf{ж)}\ Минимальные\ значения:\]

\[2\sin x = - 2\]

\[\sin x = - 1\]

\[x = - \arcsin 1 + 2\pi n\]

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

\[\textbf{з)}\ Возрастает:\]

\[- \frac{\pi}{2} + 2\pi n < x < \frac{\pi}{2} + 2\pi n;\]

\[убывает:\]

\[\frac{\pi}{2} + 2\pi n < x < \frac{3\pi}{2} + 2\pi n;\]

\[положительна:\]

\[2\pi n < x < \pi + 2\pi n;\]

\[отрицательна:\]

\[- \pi + \pi n < x < 2\pi n.\]