I'm studying for a final and came across the following question:
In a discussion with my professor, he said using Chebyshev polynomials would be messy and unwieldy and encouraged another route - he also said the hint provided in the question gives a nudge as to the path I could take, but I'm completely lost.
Chebyshev roots are proven to be the optimal places for interpolation (optimal meaning they give you the least amount of error in your interpolation). The roots of the $n$th Chebyshev Polynomial in the interval $(-1,1)$ are given by $$ z_k = \cos \left( \frac{2k-1}{2n}\right), k = 1,2,...n$$
Since you need them to be on the interval $[0,1]$ instead, take your $z_k$ and run them through the function $$T(z) = \frac{1}{2}z + \frac{1}{2} $$ to get them into the correct interval, but still in the correct places with respect to one another.
See where this takes you! Comment if you are stuck again.