2.(20%) The Cobb-Douglas Model for the production function is $P = bL^q K^{1-q}$, where $b$ and $\alpha$ are positive constants and $\alpha < 1$. We want to maximize the production $P$ with the constraint that $mL + nK = q$, where $m$, $n$ and $q$ are positive constants. Show that the maximum production occurs when $$L = \frac{\alpha q}{m}, \quad K = \frac{(1 - \alpha)q}{n}$$