Question

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

Answer 1

Yes you can

Explanation:

`frac{d}{dx}(1/f(x)) = frac{f(x)frac{d}{dx}(1)-(1)frac{d}{dx}(f(x))}{f(x)^2}`

`= -frac{f'(x)}{f(x)^2}`

Since `x > 0` and `f'(x) > 0`, we conclude that

`frac{d}{dx}(1/f(x)) = -frac{f'(x)}{f(x)^2} < 0`

for all `x>0`.

Therefore, `1/f(x)` is monotonically decreasing.