When can I say that a utility function has constant marginal utility? Does this utility function have increasing/decreasing or constant marginal utility?

$ U(x,y) = x^2 y^2 $

Now, $ f_x = 2xy^2 $, $ f_{xx} = 2y^2 $, $ f_y = 2yx^2 $, $f_{yy} = 2x^2 $

$ f_{xx} $ has no $ x $ term in it -- so is the marginal utillity of $ x$ constant or increasing? It increases as $ y $ increases, of course, but it stays constant if we increase $ x $. My textbook says that this is a case of increasing marginal utility, but I don't understand why. Similar problem for $ f_{yy} $ , which has no $ y $ term.
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.