Finding demand function given a utility min(x,y) function
No, you should not use Lagrange multipliers here, but sound thinking. Suppose $x\neq y$, say for concreteness $x&lt;y$. Let $\epsilon=y-x$. Then $\min\{x,y\}=x=\min\{x,x\}=\min\{x,y-\epsilon\}.$
So the consumer could reduce her consumption of good 2, without being worse off. On the other hand for all $\delta&gt;0$, we would have $\min\{x+\delta,y-\epsilon/2\}&gt;x=\min\{x,y\}$, so the consumer could be better of by reducing the consumption of the second good and spending the freed money on the first good. In an optimum, a consumer cannot improve so optimality requires $x=y$. It is also clear that consumers improve along the $x=y$ 45° ray. So you can simply use $x=y$ as an optimality condition to be substituted into your budget constraint and bypass Lagrange multipliers.