Question

Determine those values of $x$ for which $f(x) = \frac{2}{x} - 5$, $x \ne 0$ is increasing or decreasing.

Hint:

If the derivative $f'(x) < 0$ for all values in a domain, then the function is decreasing throughout that domain.

Answer 1

To determine if the function is increasing or decreasing, we compute its derivative. Given \( f(x) = \frac{2}{x} - 5 \), the derivative with respect to \( x \) is: \[ f'(x) = \frac{d}{dx}\left(\frac{2}{x}\right) - \frac{d}{dx}(5) = -\frac{2}{x^2} - 0 = -\frac{2}{x^2} \] Observing the sign of \( f'(x) \): For all \( x e 0 \), \( x^2>0 \), which implies \( \frac{2}{x^2}>0 \), and thus \( -\frac{2}{x^2}<0 \). Therefore, \( f'(x)<0 \) for all \( x e 0 \), indicating that the function is decreasing for all \( x e 0 \).