Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 12F / iii / Solution

Enumerate the finite set increasingly, $y_1<\cdots <y_{n+2}$, so alternating signs refer to spatial order. This is the enumeration needed for the interpolation construction below. The Lagrange cardinal polynomials satisfy $r_j(y_k)={\delta }_{jk}$ and have degree $n+1$.

If the printed indexing were interpreted as an arbitrary fixed permutation, the assertion in (a) would need this qualification. For example with $n=2$, $(y_1,y_2,y_3,y_4)=(0,1/3,1,2/3)$ gives $r(x)=-1+9x(1-x)$ of degree two; for $f=0$ every $\lambda$ works. Ordering the nodes resolves the issue and yields the requested minimizer for every set $Y$.