This question asks for the minimum value of the sum of n positive real numbers when their product is a fixed constant, denoted by C. This is a classic optimization problem that can be effectively solved using the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

Understanding the AM-GM Inequality

The AM-GM inequality states that for any set of n non-negative real numbers \(a_1, a_2, ..., a_n\), the arithmetic mean (average) is greater than or equal to the geometric mean. Mathematically, it is expressed as:

$$ \frac{a_1 + a_2 + ... + a_n}{n} \ge \sqrt[n]{a_1 \cdot a_2 \cdot ... \cdot a_n} $$

The equality holds true (i.e., the minimum value of the arithmetic mean is achieved) if and only if all the numbers in the set are equal, meaning \(a_1 = a_2 = ... = a_n\).

Applying AM-GM to Find the Minimum Sum

We are given n positive real numbers: \(a_1, a_2, a_3, ..., a_n\).

We are also given that their product is a fixed constant C:

$$ a_1 \cdot a_2 \cdot ... \cdot a_n = C $$

We need to find the minimum value of their sum, let's call it S:

$$ S = a_1 + a_2 + ... + a_n $$

Now, let's apply the AM-GM inequality:

$$ \frac{a_1 + a_2 + ... + a_n}{n} \ge \sqrt[n]{a_1 \cdot a_2 \cdot ... \cdot a_n} $$

Substitute the sum S and the product C into the inequality:

$$ \frac{S}{n} \ge \sqrt[n]{C} $$

To find the minimum value of S, we isolate S by multiplying both sides of the inequality by n:

$$ S \ge n \sqrt[n]{C} $$

This inequality shows that the sum S must be greater than or equal to \(n C^{1/n}\). Therefore, the minimum value of the sum S is \(n C^{1/n}\).

Condition for the Minimum Value

The minimum value occurs when the equality condition of the AM-GM inequality is met. This happens when all the numbers are equal:

$$ a_1 = a_2 = ... = a_n $$

Let \(a\) be the common value, so \(a_1 = a_2 = ... = a_n = a\).

Their product is \(a \cdot a \cdot ... \cdot a\) (n times), which equals \(a^n\).

Since the product is given as C:

$$ a^n = C $$

Solving for a, we get:

$$ a = C^{1/n} $$

The sum in this case is \(S = n \cdot a\):

$$ S = n \cdot C^{1/n} $$

This confirms that the minimum value of the sum is indeed \(n C^{1/n}\).

Conclusion

Based on the application of the AM-GM inequality, the minimum value of the sum of n positive real numbers whose product is a fixed number C is \(n C^{1/n}\).