I need to maximize $U = BM$ with constraits:
$6B +3M = 60$, $B>0$ and $M>0$.
The Lagrange function is $L=U + \lambda (6B+3M-60) + KB + HM$.
So
$\partial_{\lambda}L= 6B+3M-60=0$ $\partial_{K}L = B=0$ $\partial_{H}L = M=0$ $\partial_{M}L = \partial_{M} (BM) +\lambda(3+6\partial_{M}B)+K\partial_{M}B +H=((-0.5)M+B)-0.5K+H$
where $B=\frac{60-3M}{6}$ so $\partial_{M}B = -\frac{1}{2}$. But I am confused here by constrait $B=0$ and $M=0$, they cannot be. How can I make sure $B$ and $M$ are non-negative?