Past exam of the options I course of the University of Oxford / 2012 / P / P1 / c / Solution

Write $p_1(x)=a+bx$. It suffices to impose orthogonality of the error to the basis $1,x$ of ${\Pi }_1$. With the printed weight $x^{-1/2}$ and $f=x^{3/2}$, the normal equations areHere, for example, $⟨f,1⟩={\int }_0^{1}xdx=1/2$ and $⟨f,x⟩={\int }_0^1{x}^{2}dx=1/3$. Solving gives $a=-1/16$, $b=15/16$. ThereforeThe two normal equations make the error orthogonal to every linear polynomial, and part (b) then proves optimality, rather than merely furnishing a candidate fit.