If u, v, w are real numbers such that 3 3 3u 8v 27 w  18uvw, which one of the following iscorrect?

To analyze the expression $3u^3 + 8v^3 + 27w^3 - 18uvw = 0$ and determine which statement is correct, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

1. Break Down the Problem

   We start with the equation: $3u^3 + 8v^3 + 27w^3 - 18uvw = 0$ Rearranging it gives us: $3u^3 + 8v^3 + 27w^3 = 18uvw$

2. Relevant Concepts

   The AM-GM inequality states that for non-negative real numbers $a, b,$ and $c$: $\frac{a + b + c}{3} \geq \sqrt[3]{abc}$ with equality when $a = b = c$.

3. Analysis and Detail

   In our case:

   • We can set $a = 3u^3$, $b = 8v^3$, and $c = 27w^3$.
   • We find the arithmetic mean of these terms: $\frac{3u^3 + 8v^3 + 27w^3}{3} = \frac{18uvw}{3} = 6uvw$
   • Thus, we can apply the AM-GM inequality: $6uvw \geq \sqrt[3]{(3u^3)(8v^3)(27w^3)}$
   • Computing the right-hand side yields: $\sqrt[3]{3 \cdot 8 \cdot 27 \cdot u^3 \cdot v^3 \cdot w^3} = \sqrt[3]{648uvw^3}$
   • Simplifying this gives us: $6uvw \geq 6uvw$

4. Verify and Summarize

   We see that $6uvw = 6uvw$ holds, indicating equality in the AM-GM inequality. Therefore, the condition for equality must be true: $3u^3 = 8v^3 = 27w^3$ This implies that the ratios of the variables must hold:

   • From $3u^3 = 8v^3$ we can extract relations between $u$ and $v$.
   • From $3u^3 = 27w^3$ we can extract relations between $u$ and $w$.

Final Answer

Therefore, the correct conclusion is that $\frac{u}{1} = \frac{v}{\frac{3}{2}} = \frac{w}{\frac{1}{3}}$, and the values of $u, v, w$ must adhere to these proportional relationships.