Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 1033

\[1)\ y = \sqrt{\log_{0,8}\left( x^{2} - 5x + 7 \right)}\]

\[\log_{0,8}\left( x^{2} - 5x + 7 \right) \geq 0\]

\[0 < x^{2} - 5x + 7 \leq 1.\]

\[x^{2} - 5x + 7 > 0\]

\[D = 25 - 28 = - 3 < 0\]

\[x \in R.\]

\[x^{2} - 5x + 7 \leq 1\]

\[x^{2} - 5x + 6 \leq 0\]

\[D = 25 - 24 = 1\]

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

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

\[(x - 2)(x - 3) \leq 0\]

\[2 \leq x \leq 3.\]

\[Ответ:\ \ x \in \lbrack 2;\ 3\rbrack.\]

\[2)\ y = \sqrt{\log_{0,5}\left( x^{2} - 9 \right)}\]

\[\log_{0,5}\left( x^{2} - 9 \right) \geq 0\]

\[0 < x^{2} - 9 \leq 1.\]

\[1)\ x^{2} - 9 > 0\]

\[(x + 3)(x - 3) > 0\]

\[x < - 3;\ \ \ x > 3.\]

\[2)\ x^{2} - 9 \leq 1\]

\[x^{2} - 10 \leq 0\]

\[\left( x + \sqrt{10} \right)\left( x - \sqrt{10} \right) \leq 0\]

\[- \sqrt{10} \leq x \leq \sqrt{10}.\]

\[Ответ:\ \ \]

\[x \in \left\lbrack - \sqrt{10};\ - 3 \right) \cup \left( 3;\ \sqrt{10} \right\rbrack.\]

\[3)\ y = \sqrt{\log_{4}(1 + 6x) + \left| \log_{\frac{1}{8}}(1 + 7x) \right|}\]

\[\log_{4}(1 + 6x) + \left| \log_{\frac{1}{8}}(1 + 7x) \right| \geq 0\]

\[\log_{2}\sqrt{1 + 6x} + \left| \log_{2}\sqrt[3]{1 + 7x} \right| \geq 0.\]

\[\log_{2}\sqrt[3]{1 + 7x} \leq 0\]

\[0 < \sqrt[3]{1 + 7x} \leq 1\]

\[0 < 1 + 7x \leq 1\]

\[- 1 < 7x \leq 0\]

\[- \frac{1}{7} < x \leq 0.\]

\[- \frac{1}{7} < x \leq 0:\]

\[\log_{2}\sqrt{1 + 6x} - \log_{2}\sqrt[3]{1 + 7x} \geq 0\]

\[\frac{\sqrt{1 + 6x}}{\sqrt[3]{1 + 7x}} \geq 1\]

\[\sqrt{1 + 6x} \geq \sqrt[3]{1 + 7x}\]

\[(1 + 6x)^{3} \geq (1 + 7x)^{2}\]

\[216x^{3} + 108x^{2} + 18x + 1 \geq 1 + 14x + 49x^{2}\]

\[216x^{3} + 59x^{2} + 4x \geq 0\]

\[x\left( 216x^{2} + 59x + 4 \right) \geq 0\]

\[D = 3481 - 3456 = 25\]

\[x_{1} = \frac{- 59 - 5}{2 \bullet 216} = - \frac{4}{27};\]

\[x_{2} = \frac{- 59 + 5}{2 \bullet 216} = - \frac{1}{8};\]

\[\left( x + \frac{4}{27} \right)\left( x + \frac{1}{8} \right)x \geq 0\]

\[- \frac{4}{27} \leq x \leq - \frac{1}{8};\ \ \ x \geq 0.\]

\[x \geq 0:\]

\[\log_{2}\sqrt{1 + 6x} + \log_{2}\sqrt[3]{1 + 7x} \geq 0\]

\[\sqrt{1 + 6x} \bullet \sqrt[3]{1 + 7x} \geq 1\]

\[(1 + 6x)^{3} \bullet (1 + 7x)^{2} \geq 1\]

\[x \in R.\]

\[Ответ:\ \ \]

\[x \in \left( - \frac{1}{7};\ - \frac{1}{8} \right\rbrack \cup \lbrack 0;\ + \infty).\]

\[4)\ y = \sqrt{\left| \log_{27}\left( 1 + \frac{7}{2}x \right) \right| - \log_{\frac{1}{3}}(1 + 2x)}\]

\[\left| \log_{27}\left( 1 + \frac{7}{2}x \right) \right| - \log_{\frac{1}{3}}(1 + 2x) \geq 0\]

\[\left| \log_{3}\sqrt[3]{1 + 3,5x} \right| + \log_{3}(1 + 2x) \geq 0.\]

\[\log_{3}\sqrt[3]{1 + 3,5x} \leq 0\]

\[0 < \sqrt[3]{1 + 3,5x} \leq 1\]

\[0 < 1 + 3,5x \leq 1\]

\[- 1 < 3,5x \leq 0\]

\[- \frac{2}{7} < x \leq 0.\]

\[- \frac{2}{7} < x \leq 0:\]

\[- \log_{3}\sqrt[3]{1 + 3,5x} + \log_{3}(1 + 2x) \geq 0;\]

\[\frac{1 + 2x}{\sqrt[3]{1 + 3,5x}} \geq 1;\]

\[1 + 2x \geq \sqrt[3]{1 + 3,5x}\]

\[(1 + 2x)^{3} \geq 1 + 3,5x\]

\[8x^{3} + 12x^{2} + 6x + 1 \geq 1 + 3,5x\]

\[8x^{3} + 12x^{2} + 2,5x \geq 0\]

\[\frac{1}{2}x\left( 16x^{2} + 24x + 5 \right) \geq 0\]

\[D = 576 - 320 = 256\]

\[x_{1} = \frac{- 24 - 16}{2 \bullet 16} = - \frac{5}{4};\]

\[x_{2} = \frac{- 24 + 16}{2 \bullet 16} = - \frac{1}{4};\]

\[\left( x + \frac{5}{4} \right)\left( x + \frac{1}{4} \right)x \geq 0\]

\[- \frac{5}{5} \leq x \leq - \frac{1}{4};\ \ x \geq 0.\]

\[x \geq 0:\]

\[\log_{3}\sqrt[3]{1 + 3,5x} + \log_{3}(1 + 2x) \geq 0\]

\[\sqrt[3]{1 + 3,5x} \bullet (1 + 2x) \geq 1\]

\[(1 + 3,5x) \bullet (1 + 2x)^{3} \geq 1\]

\[x \in R.\]

\[Ответ:\ \ \]

\[x \in \left( - \frac{2}{7};\ - \frac{1}{4} \right\rbrack \cup \lbrack 0;\ + \infty).\]