analysis problem, details please...

Problem 9 [10] Suppose f is a real-valued differentiable function and f' E C([0, 1]) and f(0) = 0, prove that, VE > 0, there is a polynomials p of the form p(x) = _- ant12nt such that Ilp - fll + Ilp' - f'llse, where g | = sup xEjo,1 9(x), Vg E C([0, 1])