# How do you use the quotient rule to show that 1/f(x) is decreasing given that f(x) is a positive increasing function defined for all x?

Apr 23, 2015

Assuming that $f \left(x\right) > 0$ and $f \left(x\right)$ is increasing i. e. $f ' \left(x\right) > 0$.

Let $g \left(x\right) = \frac{1}{f} \left(x\right)$. Then:

$g ' \left(x\right) = \left(\frac{1}{f} \left(x\right)\right) '$
According to the quotient rule $g ' \left(x\right)$ can be calculated as:

$g ' \left(x\right) = \frac{1 ' \cdot f \left(x\right) - 1 \cdot f ' \left(x\right)}{{f}^{2} \left(x\right)} = \frac{0 \cdot f \left(x\right) - 1 \cdot f ' \left(x\right)}{{f}^{2} \left(x\right)} = \frac{- f ' \left(x\right)}{f} ^ 2 \left(x\right) < 0$

$g ' \left(x\right) < 0$ indicates that $g \left(x\right)$ is decreasing.