Finding the maximum without calculus Given that $x+y=6$. Is there any way to find the maximum value of $x^2y$ without using calculus or graphical method? 

Does not exist! Try $y\rightarrow+\infty$.
For non-negative variables we can use AM-GM:
$$x^2y=\frac{1}{2}x\cdot x\cdot2y\leq\frac{1}{2}\left(\frac{x+x+2y}{3}\right)^3=32.$$
The equality occurs for $x=4$ and $y=2$, which says that $32$ is a maximal value.
