Can a monopoly supply negative units to one of its two markets?
Marginal cost is only guaranteed to be equal to marginal revenue in optima where $Y &gt; 0$ (and when the profit function is concave, but in your case it is). If 
$$
MC(0) &gt; MR(0)
$$
the producing the first unit results in a loss, and a profit maximizing monopolist would not produce at all (if the profit function is concave).

An example: I am willing to buy golf ball sized diamonds from you. I am willing to pay 5 dollars for the first, 4 dollars for the second, 3 dollars for the third and so on, ending in 0 dollars for the sixth and any further units. How many golf ball sized diamonds will you sell me?