Given that $x+y=6$. Is there any way to find the maximum value of $x^2y$ without using calculus or graphical method?
2026-03-30 22:52:58.1774911178
Finding the maximum without calculus
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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.