<s>[INST] Suppose $x, y, z$ are positive real number such that $x + 2y + 3z = 1$. Find the maximum value of $xyz^2$ I'm trying to solve this problem using Lagrange's multiplier method but I'm unable to get the value of lambda and also $z.$ kindly help  me out with this problem.
I'm getting $x = 2y$
I'm stuck somewhere between the steps 
 [/INST] Alt. hint (no calculus):   the following inequality holds by AM-GM, with equality iff $\displaystyle x = 2y = \frac{3}{2}z\,$:
$$
\frac{\sqrt{3}}{\sqrt[4]{2}} \cdot \sqrt[4]{xyz^2} = \sqrt[4]{x \cdot 2y \cdot \frac{3}{2}z \cdot \frac{3}{2}z} \;\le\; \frac{x+2y+3z}{4} = \frac{1}{4} \;\;\iff\;\;xyz^2 \le \frac{2}{3^2 \cdot 4^4} = \frac{1}{1152}
$$
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