Let $f:(0,\infty ) \rightarrow (0,\infty )$ be strictly decreasing function. Consider
$$ h(x) = \frac{f(\frac{x}{1+x})}{1+f(\frac{x}{1+x})} $$
Options:
$(A)$ $h$ is strictly decreasing. $(B)$ $h$ is strictly increasing. $(C)$ $h$ is strictly decreasing at first and then strictly increasing. $(D)$ $h$ is strictly increasing at first and then strictly decreasing .
My attempt:
Strictly increasing (decreasing) nature of a function depends on the derivative hence I took the derivative of $h$ using the definition provided as follows, $$h'(x) = \frac{f'(.)}{(1+x)^2} \frac{1}{(1+f(.)))^2} $$
Now, denominator is positive quantity and numerator is negative (as $f$ is strictly decreasing) So, we can conclude h is strictly decreasing.
Is this approach correct? any improvements would be appreciated...