(a) State and prove the Lagrangian sufficiency theorem.

(b) Let $n \geqslant 1$ be a given constant, and consider the problem:

$$\operatorname{minimise} \sum_{i=1}^{n}\left(2 y_{i}^{2}+x_{i}^{2}\right) \text { subject to } x_{i}=1+\sum_{k=1}^{i} y_{k} \text { for all } i=1, \ldots, n$$

Find, with proof, constants $a, b, A, B$ such that the optimal solution is given by

$$x_{i}=a 2^{i}+b 2^{-i} \text { and } y_{i}=A 2^{i}+B 2^{-i} \text {, for all } i=1, \ldots, n .$$