Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 220

\[1)\ f(x) = x - \ln x;\]

\[f^{'}(x) = 1 - \frac{1}{x} > 0;\]

\[\frac{x - 1}{x} > 0\]

\[x < 0;\text{\ \ \ x} > 1.\]

\[Ответ:\ \ \]

\[f^{'}(x) = 0\ при\ x = 1;\]

\[f^{'}(x) > 0\ при\ x > 1;\]

\[f^{'}(x) < 0\ при\ 0 < x < 1.\]

\[2)\ f(x) = x\ln x;\]

\[f^{'}(x) = \ln x + x \bullet \frac{1}{x} > 0;\]

\[\ln x + 1 > 0\]

\[\ln x > - 1\]

\[x > e^{- 1}.\]

\[Ответ:\ \ \]

\[f^{'}(x) = 0\ при\ x = e^{- 1};\]

\[f^{'}(x) > 0\ при\ x > e^{- 1};\]

\[f^{'}(x) < 0\ при\ 0 < x < e^{- 1}.\]

\[3)\ f(x) = x^{2}\ln x;\]

\[f^{'}(x) = 2x \bullet \ln x + x^{2} \bullet \frac{1}{x} > 0;\]

\[2x\ln x + x > 0\]

\[x\left( 2\ln x + 1 \right) > 0\]

\[2\ln x > - 1\]

\[\ln x > - \frac{1}{2}\]

\[x > e^{- \frac{1}{2}}.\]

\[Ответ:\ \ \]

\[f^{'}(x) = 0\ при\ x = e^{- \frac{1}{2}};\]

\[f^{'}(x) > 0\ при\ x > e^{- \frac{1}{2}};\]

\[f^{'}(x) < 0\ при\ 0 < x < e^{- \frac{1}{2}}.\]

\[4)\ f(x) = x^{3} - 3\ln x;\]

\[f^{'}(x) = 3x^{2} - 3 \bullet \frac{1}{x} > 0;\]

\[3x^{3} - 3 > 0\]

\[3x^{3} > 3\]

\[x^{3} > 1\]

\[x > 1.\]

\[Ответ:\ \ \]

\[f^{'}(x) = 0\ при\ x = 1;\]

\[f^{'}(x) > 0\ при\ x > 1;\]

\[f^{'}(x) < 0\ при\ 0 < x < 1.\]