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 analyze whether the function is increasing or decreasing, we calculate the derivative of $f(x)$. Given: \[ f(x) = \frac{2}{x} - 5 \] Differentiate $f(x)$ with respect to $x$: \[ f'(x) = \frac{d}{dx}\left(\frac{2}{x}\right) - \frac{d}{dx}(5) = -\frac{2}{x^2} - 0 = -\frac{2}{x^2} \] Now observe the sign of $f'(x)$: - For all $x \ne 0$, $x^2 > 0$ ⟹ $\frac{2}{x^2} > 0$ ⟹ $-\frac{2}{x^2} < 0$ So, $f'(x) < 0$ for all $x \ne 0$. This implies the function is decreasing for all $x \ne 0$.