If the product of \(n\) positive numbers is \(n^n\), then what is the minimum value of their average for \(n = 6\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 6
For positive numbers, the arithmetic mean (average) is never less than the geometric mean, and the two are equal only when all the numbers are equal (AM ≥ GM). The geometric mean of \(n\) numbers is the \(n\)th root of their product.
Product of the 6 numbers \(=6^6\)
Geometric mean \(=\sqrt[6]{6^6}=6\)
So the average is at least 6, and it equals 6 when all six numbers are 6 (\(6\times6\times6\times6\times6\times6=6^6\))