Q:

Show that a function has an inverse.

Let $f : \mathbb{R}  \to \mathbb{R}$
defined such that  $ f = \frac{6}{7}x^{7} -3x^4 +6x -5 $ 
I need to show that $f$ has an inverse.
Well, $f$ is obviously continuous and differentiable because $f$ is just a sum of some polynomails which are known as continuous and differentiable functions.
Now, $f$ has an inverse if and only if shes one to one and onto. 
It is enough to show that $f$ is one to one by showing that $f' > 0 \; \;  \forall x \in \mathbb{R}$
However, I find it rather difficult showing that $f' = 6x^6 -12x^3 + 6 >0 \; \;  \forall x \in \mathbb{R}$
How should one be dealing with such question?

A:

Since$$(\forall x\in\mathbb R):f'(x)=6(x^3-1)^2,$$you only have $f'(x)=0$ when $x=1$; otherwise, $f'(x)>0$. Can you take it from here?

