By Weierstrass Approximation theorem, a continuous function on the interval $[0,1]$ can be uniformly approximated by polynomials. But if the function $f$ is continuously differentiable on $[0,1]$, then we can actually say something about the rate of approximation. That is, if $\epsilon > 0$, then $f$ can be approximated uniformly within $\epsilon$ by a polynomial of degree not greater than $N = N(\epsilon)$. How can I find $N(\epsilon)$?
It is trivial that $f'$ can be approximated uniformly by polynomials since it is continuous in $[0,1]$ too, but I don't know whether this has something to do with solving the problem.