$x^n/(1+x^n) $is monotonic

Question

$x^n/(1+x^n)$ is monotonic , $x \in R^+ $ , $n \in N$ tried solving it by differentiating but not getting feasible result, also is there any other way of computation through which this can be solved easily?

any hint would be appreciated

Answer 1

For all $n \in \mathbb {N}$ and $x \in \mathbb {R}_{+}$, $$ \begin{align} \frac {\text {d}}{\text {d} x} \frac {{x}^{n}}{{x}^{n} + 1} & = \frac {\text {d}}{\text {d} x} \Big( 1 - \frac {1}{{x}^{n} + 1} \Big) \\ & = \frac {1}{{\left( {x}^{n} + 1 \right)}^{2}} \cdot \frac {\text {d}}{\text {d} x} {x}^{n} = \frac {n {x}^{n - 1}}{{\left( {x}^{n} + 1 \right)}^{2}} > 0. \end{align} $$ So the function is monotonic.

Answer 2

A composition of monotonic functions is monotonic (even if one is increasing and the other is decreasing). Here the two functions are $$x\mapsto x^n$$ and $$x\mapsto \frac{x}{1+x}$$