When can I say that a utility function has constant marginal utility?
Marginal utility (of $x$) in your case is $U_x(x,y)=2xy^2$. You use the sign of the derivative of MU, namely $U_{xx}$, to tell whether MU is increasing, constant, or decreasing. 

Specifically, you have 


increasing MU if $U_{xx}&gt;0$,
constant MU if $U_{xx}=0$, and 
decreasing MU if $U_{xx}&lt;0$.


In your case, assuming that $y&gt;0$, you'd have $U_{xx}&gt;0$, hence increasing MU.