A Taylor expansion inequality

Question

I would like to show that for all $x \in [0,1]$, $$ (1+x)^n - (1-x)^n \leq 2nx (1+x)^{n-1}. $$ The rough intuition I have is that $(1+x)^n \approx 1+xn$ and $(1-x)^n \approx 1-xn$ but then I'm missing the entire $(1+x)^{n-1}$ term.

Answer 1

For $n\geq1$.

Define the function $f(x)=(1+x)^n$.

Apply the mean value theorem on the interval $[-x,x]$.

$f(x)$ is continuous and differentiable on that interval.

$(1+x)^n-(1-x)^n=2n(1+c)^{n-1} x$ for some c $\in(-x,x)$

But then $(1+c)^{n-1}\leq(1+x)^{n-1}$

the result follows.