Does $U(x,y) = x^2 + y^2 + 2xy$ represent transitive, monotonic preferences? I'm a monitor for a microeconomics course and a student came up with this question. That this utility function represents monotonic preferences I think it's clear. Both goods have positive and constant marginal utilities. What I think is less clear is if this preference relation is transitive. How can I assess it?
Any preference relation represented by an utility function is transtive.

Suppose $x \succsim y$ and  $y \succsim z$ and $\succsim$ is represented by $U$. Then $U(x) \geq U(y)  $ and $U(y) \geq U(z)$, so $U(x) \geq U(z)$  and then $x \succsim z$.