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Suppose we have the production function $f: \mathbb{R}^{2} \to \mathbb{R}$ given by $$ f(x,y) = ax + by $$ and input prices $p_{1}$ and $p_{2}$, and we want to minimize the cost function $p_{1}x_{1} + p_{2}x_{2}$ such that $ax + by \geq \alpha$ where $\alpha \in \mathbb{R}$ is some production level.

Taking the Lagrangian $$ \mathcal{L} = p_{1}x_{1} + p_{2}x_{2} + \lambda(\alpha - ax_{1} - bx_{2}) $$ and the Kuhn-Tucker first order conditions are $p_{1} - \lambda a \geq 0$ and $x_{1} \geq 0$ (but both can't be non-zero), $p_{2} - \lambda b \geq 0$ and $x_{2} \geq 0$ (without both being non-zero) and $\alpha - a x_{1} - bx_{2} \leq 0$ and $\lambda \geq 0$ (but both can't be non-zero).

How would we divide up our cases here to check whether we get a corner or interior solution? It's easy to see that the constraint has to be binding since we're minimizing cost and our objective function is strictly increasing in $x_{i}$, but how do we check the cases whether $x_{i} = 0$ or $x_{i} > 0$ for $i = 1$ or $i=2$ (or both).

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