Short cuts to solve Cobb Douglas Utility function (minimization) Say a Cobb Douglas like:

$$\max_{X,Y\: s.t. X \cdot P_x+ Y \cdot P_y=I} U=X^\alpha Y^\beta$$

When it comes to maximization I would do the following way (for the fastest result):

x: $\alpha/(\alpha + \beta) = r; r*income = r_2$; $r_2$/price good = x$

Is there a similar (or really fast) method to solve for minimization problems?
A shortcut to solve $$\min_{x,y} x^{\alpha} y^{\beta}$$ s.t. $$xp_x + yp_x \leq m$$ and $$0 \leq x,y$$ is to set $x = 0$ and/or $y = 0$, as this results in $U(x,y) = 0$, and the function does not map to negative values.