State and prove the Lagrangian Sufficiency Theorem.

The manufacturers, $A$ and $B$, of two competing soap powders must plan how to allocate their advertising resources ( $X$ and $Y$ pounds respectively) among $n$ distinct geographical regions. If $x_{i} \geqslant 0$ and $y_{i} \geqslant 0$ denote, respectively, the resources allocated to area $i$ by $A$ and $B$ then the number of packets sold by $A$ and $B$ in area $i$ are

$$\frac{x_{i} u_{i}}{x_{i}+y_{i}}, \quad \frac{y_{i} u_{i}}{x_{i}+y_{i}}$$

respectively, where $u_{i}$ is the total market in area $i$, and $u_{1}, u_{2}, \ldots, u_{n}$ are known constants. The difference between the amount sold by $A$ and $B$ is then

$$\sum_{i=1}^{n} \frac{x_{i}-y_{i}}{x_{i}+y_{i}} u_{i}$$

$A$ seeks to maximize this quantity, while $B$ seeks to minimize it.

(i) If $A$ knows $B$ 's allocation, how should $A$ choose $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ ?

(ii) Determine the best strategies for $A$ and $B$ if each assumes the other will know its strategy and react optimally.