Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 285

\[1)\ f(x) = x^{2} + \frac{16}{x^{2}}\ при\ x > 0:\]

\[f^{'}(x) = 2x + 16 \bullet \left( - \frac{2}{x^{3}} \right) =\]

\[= 2x - \frac{32}{x^{3}} = \frac{2x^{4} - 32}{x^{3}}.\]

\[2x^{4} - 32 \geq 0\]

\[x^{4} - 16 \geq 0\]

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

\[(x + 2)(x - 2) \geq 0\]

\[x \leq - 2;\text{\ \ \ x} \geq 2.\]

\[f(2) = 4 + \frac{16}{4} = 4 + 4 = 8.\]

\[Ответ:\ \ 8.\]

\[2)\ f(x) = x + \frac{4}{x}\ при\ x > 0:\]

\[f^{'}(x) = 1 + 4 \bullet \left( - \frac{1}{x^{2}} \right) = \frac{x^{2} - 4}{4x^{2}}.\]

\[x^{2} - 4 \geq 0\]

\[(x + 2)(x - 2) \geq 0\]

\[x \leq - 2;\text{\ \ \ x} \geq 2.\]

\[f(2) = 2 + \frac{4}{2} = 2 + 2 = 4.\]

\[Ответ:\ \ 4.\]