Prove: If the function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f'(x) = 0$ on $(a,b)$, then $f$ must be a constant function on $[a,b]$.
I need to select some $x_1$ and $x_2$ in $[a,b]$ such that $x_1$ is not equal to $x_2$ therefore by the mean value theorem.. This is where I am getting a bit confused how to apply MVT to this proof.. Hints/Tips appreciated. I may be over thinking things. Thanks.