# Strictly monotonic utility function

Given a utility function of $$U = x - 3y^2$$ for $$x>0$$ and $$y>0$$

Are the preferences strictly monotonic for all $$x>0$$ and $$y>0$$? what happens to the marginal utility as each good is being consumed more?

I was able to get $$U_1 = 1$$ AND $$U_2= -9y$$ so is it strictly monotonic because $$U_1$$ is a constant?

You are only looking for $$x>0$$ and $$y>0$$. There you have
• $$U_x = 1 > 0 \implies U$$ is increasing in $$x$$ for all $$x>0$$, and
• $$U_y = -6y < 0 \implies U$$ is decreasing in $$y$$ for all $$y>0$$
This, indeed, $$U$$ is monotonic over $$x>0,y>0$$, increasing in $$x$$ and decreasing in $$y$$.
According to the usual defintion of "strict monotonicity" of Utility functions the function also has to be non decreasing for all arguments. This is not the case here since you loose Utility if you consume more from good $$y$$