The product of n positive numbers is unity. T | vedclass

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(D) Given that the product of $n$ positive numbers $x_1, x_2, \dots, x_n$ is $1$,i.e.,$x_1 \cdot x_2 \cdot \dots \cdot x_n = 1$. By the Arithmetic Mea

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Never less than $n$

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(D) Given that the product of $n$ positive numbers $x_1, x_2, \dots, x_n$ is $1$,i.e.,$x_1 \cdot x_2 \cdot \dots \cdot x_n = 1$. By the Arithmetic Mean-Geometric Mean $(AM-GM)$ inequality,we know that for positive real numbers,the Arithmetic Mean is always greater than or equal to the Geometric Mean: $\frac{x_1 + x_2 + \dots + x_n}{n} \ge \sqrt[n]{x_1 \cdot x_2 \cdot \dots \cdot x_n}$ Substituting the given product: $\frac{x_1 + x_2 + \dots + x_n}{n} \ge \sqrt[n]{1} = 1$ Multiplying both sides by $n$: $x_1 + x_2 + \dots + x_n \ge n$ Therefore,the sum of the $n$ positive numbers is never less than $n$.

The product of $n$ positive numbers is unity. Their sum is

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