Решебник по алгебре и начала математического анализа 11 класс Алимов Задание 730

\[\boxed{\mathbf{730}\mathbf{.}}\]

\[y = \sin x;\]

\[1)\left\lbrack \frac{\pi}{6};\ \pi \right\rbrack:\]

\[возрастает\ на\ \left\lbrack \frac{\pi}{6};\ \frac{\pi}{2} \right\rbrack;\]

\[убывает\ на\left\lbrack \frac{\pi}{2};\ \pi \right\rbrack;\]

\[y_{\min} = \sin\frac{\pi}{6} = \frac{1}{2};\]

\[y_{\min} = \sin\pi = 0;\]

\[y_{\max} = \sin\frac{\pi}{2} = 1.\]

\[Ответ:\ \ E(y) = \lbrack 0;\ 1\rbrack.\]

\[2)\left\lbrack \frac{3\pi}{4};\ \frac{5\pi}{4} \right\rbrack:\]

\[y_{\min} = \sin\frac{5\pi}{4} = \sin\left( \pi + \frac{\pi}{4} \right) =\]

\[= - \sin\frac{\pi}{4} = - \frac{\sqrt{2}}{2}\]

\[y_{\max} = \sin\frac{3\pi}{4} = \sin\left( \pi - \frac{\pi}{4} \right) =\]

\[= \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}.\]

\[Ответ:\ \ E(y) = \left\lbrack - \frac{\sqrt{2}}{2};\ \frac{\sqrt{2}}{2} \right\rbrack.\]