$ x\in \left[0,{1\over n-1} \right] \to 1+nx \le (1+x)^n \le {1+x\over 1-(n-1)x}$ (Homework assignement) This is about a homework I have to do. I don't want the straight answer, just the hint that may help me start on this. To give you context, we're now studying integrals.
Now here is the question :

Prove : $ x\in \left[0,{1\over n-1} \right] \to 1+nx \le (1+x)^n \le {1+x\over 1-(n-1)x}$

The exercice suggest using what I can only poorly translate to "Inequality of finite increasing" and that states :

Let $f$ be a function continuous on $[a,b], a<b$ and differentiable on $[a,b]$.
$\exists M \in \mathbb{R}, \forall x \in [a,b], f'(x)\le M \to f(b)-f(a) \le M(b-a)$

I tried to apply this to $f(x)=(1+x)^n$ but to no avail.
Any input will be greatly apreciated, Thanks !

(Partial Hint)
Hint: For the first inequality, expanding the bracket using the binomial theorem is sufficient.
