# f'(x)>0 for all x. Can you use the quotient rule to show that 1/f(x) is monotonically decreasing?

Feb 14, 2016

Yes you can

#### Explanation:

$\frac{d}{\mathrm{dx}} \left(\frac{1}{f} \left(x\right)\right) = \frac{f \left(x\right) \frac{d}{\mathrm{dx}} \left(1\right) - \left(1\right) \frac{d}{\mathrm{dx}} \left(f \left(x\right)\right)}{f {\left(x\right)}^{2}}$

$= - \frac{f ' \left(x\right)}{f {\left(x\right)}^{2}}$

Since $x > 0$ and $f ' \left(x\right) > 0$, we conclude that

$\frac{d}{\mathrm{dx}} \left(\frac{1}{f} \left(x\right)\right) = - \frac{f ' \left(x\right)}{f {\left(x\right)}^{2}} < 0$

for all $x > 0$.

Therefore, $\frac{1}{f} \left(x\right)$ is monotonically decreasing.