Question

Let \( a, b, c \) be positive numbers such that \( abc = 1 \). Then the minimum value of \( a + b + c \) is:

• A. 8
• B. 4
• C. 6
• D. 2
• E. 3

Hint:

The Arithmetic Mean-Geometric Mean (AM-GM) inequality is a powerful tool for finding the minimum or maximum value of a sum under certain conditions. In this case, it helps us determine the minimum value of \( a + b + c \) when \( abc = 1 \).

Answer 1

Answer: (E)

We are given that \( abc = 1 \) and we need to find the minimum value of \( a + b + c \), where \( a, b, c \) are positive numbers.
Step 1: Use the Arithmetic Mean-Geometric Mean (AM-GM) inequality. The AM-GM inequality states that for positive numbers \( x_1, x_2, \dots, x_n \), \[ \frac{x_1 + x_2 + \dots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \dots x_n}, \] with equality holding if and only if \( x_1 = x_2 = \dots = x_n \).
Step 2: Apply the AM-GM inequality to the numbers \( a, b, c \): \[ \frac{a + b + c}{3} \geq \sqrt[3]{abc}. \] Since \( abc = 1 \), we have: \[ \frac{a + b + c}{3} \geq \sqrt[3]{1} = 1. \] Multiplying both sides by 3: \[ a + b + c \geq 3. \] Step 3: The equality holds when \( a = b = c \). Since \( abc = 1 \), if \( a = b = c \), then \( a^3 = 1 \), so \( a = 1 \). Thus, \( a = b = c = 1 \).
Therefore, the minimum value of \( a + b + c \) is \( 1 + 1 + 1 = 3 \).
Thus, the correct answer is option (E).