<p>The greatest value of \(\dfrac{xyz}{(x-m)(y-n)(z-r)}\) is</p>
Step-by-Step Solution
Key Concept: Use AM-GM inequality on the denominators after recognizing that the expression is maximized when variables are positioned symmetrically relative to their respective shifts. The critical insight is setting partial derivatives to zero or applying weighted AM-GM to find that maximum occurs at specific ratio relationships.
<p><strong>Step 1:</strong> Let u = x-m, v = y-n, w = z-r, so x = u+m, y = v+n, z = w+r where u, v, w > 0</p><p><strong>Step 2:</strong> The expression becomes $\frac{(u+m)(v+n)(w+r)}{uvw}$</p><p><strong>Step 3:</strong> Expand: $\frac{uvw + munv + nwu + rwv + mnw + nru + mwr + mnr}{uvw}$</p><p><strong>Step 4:</strong> This equals $1 + \frac{m}{u} + \frac{n}{v} + \frac{r}{w} + \frac{mn}{uv} + \frac{nr}{vw} + \frac{mr}{uw} + \frac{mnr}{uvw}$</p><p><strong>Step 5:</strong> By AM-GM inequality applied to three terms: $\frac{m}{u} + \frac{n}{v} + \frac{r}{w} \geq 3\sqrt[3]{\frac{mnr}{uvw}}$</p><p><strong>Step 6:</strong> Maximum occurs when $\frac{m}{u} = \frac{n}{v} = \frac{r}{w}$, giving the greatest value as $\boxed{\frac{(m+n+r)^3}{mnr}}$</p><p>∴ Answer: A</p>
Correct Answer: A