# 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.
• @Mathxx It means $x^2y\rightarrow+\infty$ for $y\rightarrow+\infty$. – Michael Rozenberg Aug 11 '17 at 9:16
• He means that you need to add that $x$ and $y$ are non negative to your hypothesis. – Joaquin San Aug 11 '17 at 9:35
• @Mathxx Substitute $x=4$ and $y=2$: $x+y=6$ and $x^2y=4^2\cdot2=32$. – Michael Rozenberg Aug 11 '17 at 11:47