Solution
= Solution
By part (c), the best degree-at-most-two <polynomial> is odd, so it must be $p_2(x)=bx$. <Orthogonality> to $x$ gives
$$
b=\frac{(x^{2011},x)}{(x,x)}
=\frac{\int_{-1}^1x^{2014}\,dx}{\int_{-1}^1x^4\,dx}
=\frac{2/2015}{2/5}=\frac1{403}.
$$
Therefore
$$
\boxed{p_2(x)=\frac{x}{403}.}
$$
Its error is automatically <orthogonal> to the even <basis> functions $1,x^2$ by parity, and the chosen coefficient makes it <orthogonal> to $x$, so part (b) verifies optimality.