ГДЗ по алгебре и начала математического анализа 11 класс Колягин Задание 9

\[1)\ y = \sin^{4}x + \cos^{4}x =\]

\[= 1 - \frac{1}{4} + \frac{1}{4}\cos{4x} =\]

\[= \frac{3}{4} + \frac{1}{4}\cos{4x}.\]

\[- 1 \leq \cos{4x} \leq 1;\]

\[- \frac{1}{4} \leq \frac{1}{4}\cos{4x} \leq \frac{1}{4};\]

\[\frac{1}{2} \leq \frac{3}{4} + \frac{1}{4}\cos{4x} \leq 1;\]

\[E(y) = \left\lbrack \frac{1}{2};\ 1 \right\rbrack.\]

\[2)\ y = \sin^{6}x + \cos^{6}x =\]

\[= 1 - \frac{3}{4}\sin^{2}{2x} = 1 - \frac{3}{8}\left( 1 - \cos{4x} \right) =\]

\[= 1 - \frac{3}{8} + \frac{3}{8}\cos{4x} =\]

\[= \frac{5}{8} + \frac{3}{8}\cos{4x}.\]

\[- 1 \leq \cos{4x} \leq 1;\]

\[- \frac{3}{8} \leq \frac{3}{8}\cos{4x} \leq \frac{3}{8};\]

\[\frac{1}{4} \leq \frac{5}{8} + \frac{3}{8}\cos{4x} \leq 1;\]

\[E(y) = \left\lbrack \frac{1}{4};\ 1 \right\rbrack.\]