Question

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?

Answer 1

Assuming that `f(x)>0` and `f(x)` is increasing i. e. `f'(x)>0`.

Let `g(x)=1/f(x)`. Then:

`g'(x)=(1/f(x))'`
According to the quotient rule `g'(x)` can be calculated as:

`g'(x)=(1'*f(x)-1*f'(x))/(f^2(x))=(0*f(x)-1*f'(x))/(f^2(x))=(-f'(x))/f^2(x)<0`

`g'(x) <0` indicates that `g(x)` is decreasing.