How to prove $x^3$ is strictly increasing I am trying to use $f(x)=x^3$ as a counterexample to the following statement. 
If $f(x)$ is strictly increasing over $[a,b]$ then for any $x\in (a,b), f'(x)>0$. 
But how can I show that $f(x)=x^3$ is strictly increasing?

$$\frac{d x^3}{dx}=3x^2\geqslant0, \forall x \in[1,2]$$
Function is strictly increasing in $[1,2]$ and...
