# Kuhn-Tucker conditions in linear cost minimization

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).