Past exam of the approximation theory and finite element methods course of the University of Oxford / 2009 / A / 2 / d / Solution

The best approximation in the $L^2$ norm is the orthogonal projection onto $\mathrm{span}\{{\varphi }_0,{\varphi }_1\}$. ThereforeUsing ${\int }_0^1{x}^{k}dx=1/(k+1)$,It follows thatThe error is orthogonal to every linear polynomial, so the Pythagorean identity proves optimality and uniqueness. At $k=0$, this gives $p_1=1$; at $k=1$, it gives $p_1=x$. Both are exact because the target itself then belongs to $P_1$.