ГДЗ по алгебре 11 класс Никольский Параграф 4. Производная Задание 50

\[\boxed{\mathbf{50}\mathbf{.}}\]

\[y = \frac{\ln x}{x};\ \ x > 0\]

\[y^{'} = \frac{\left( \ln x \right)^{'} \cdot x - \ln x \cdot x^{'}}{x^{2}} =\]

\[= \frac{\frac{1}{x} \cdot x - \ln x}{x^{2}} = \frac{1 - \ln x}{x^{2}}.\]

\[\textbf{а)}\ \frac{1 - \ln x}{x^{2}} = 0;x \neq 0\]

\[1 - \ln x = 0\]

\[\ln x = 1\]

\[x = e.\]

\[\textbf{б)}\ \frac{1 - \ln x}{x^{2}} > 0\]

\[1 - \ln x > 0\]

\[\ln x < 1\]

\[x < e.\]

\[x \in (0;e).\]

\[\textbf{в)}\ \frac{1 - \ln x}{x^{2}} < 0\]

\[1 - \ln x < 0\]

\[\ln x > 1\]

\[x > e.\]

\[x \in (e; + \infty).\]