5) f(x) = -5/x - 10x. Найдите 1) f'(x), 2) f'(x)=0.

Решение:

1. Найдем \( f'(x) \):

Функция: \( f(x) = -5x^{-1} - 10x \).

Производная:

\( f'(x) = \frac{d}{dx}(-5x^{-1}) - \frac{d}{dx}(10x) \)

\( f'(x) = -5 \cdot (-1)x^{-1-1} - 10 \)

\( f'(x) = 5x^{-2} - 10 \)

\( f'(x) = \frac{5}{x^2} - 10 \)

2. Решим уравнение \( f'(x) = 0 \):

\( \frac{5}{x^2} - 10 = 0 \)

\( \frac{5}{x^2} = 10 \)

\( 5 = 10x^2 \)

\( x^2 = \frac{5}{10} = \frac{1}{2} \)

\( x = \pm \sqrt{\frac{1}{2}} = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \)

Ответ: 1) \( f'(x) = \frac{5}{x^2} - 10 \); 2) \( x = \frac{\sqrt{2}}{2} \) и \( x = -\frac{\sqrt{2}}{2} \).