Consider the Leontief production function $f(x_1, x_2) = \sqrt{\min(x_1, x_2)}, x_1, x_2 \geq 0$. Then when the profit function $\Pi(p) = max_{x_1, x_2 \geq 0} = -(p_1x_1+p_2x_2) + p_3 f(x_1, x_2) $ is differentiable, i.e. any constraints for $p = (p_1, p_2, p_3)$?
We know $\Pi(p) = \frac{p_3}{4(p_1 + p_2)}$, do we just need the condition that $p_1\neq 0$ or $p_2 \neq 0$ to make sure $\Pi(p)$ is differentiable? But According to the duality theorem, $\Pi(p)$ is differentiable if $\arg \max_{x_1, x_2 \geq 0} -(p_1x_1+p_2x_2) + p_3 f(x_1, x_2) $ is unique. I have no idea how to determine the all feasible $p$ such that the maximizer is unique.