Proving Bernoulli's inequality for $x>-2$

Question

For the generalization of Bernoulli's Inequality: $(1+x)^n \geq 1+nx$ I am having trouble finding out how to prove it for when $x \geq -2$ and $n$ is in $\mathbb{N}$.

I understand how to prove the inequality for the case when $x \geq -1$ but these methods don't hold for the above case. How would I prove this?

Answer 1

HINT:

Let $f(x)=(1+x)^n-(1+nx)$.

Note that $f(-2)=2n-1-(-1)^n\ge 0$, and $f(-1)=n-1\ge 0$, for $n\ge 1$.

Furthermore, $f'(x)=n\left((1+x)^{n-1}-1\right) \le 0$ for $-2\le x\le -1$ and $n\ge 1$ and $f'(x)\ne 0$ for $x\in (-2,-1)$.