Question

Let \( a, b, c \) be positive numbers. If \( a + b + c \geq K \left[ (a + b)(b + c)(c + a) \right]^{1/3} \), then the maximum value of \( K \) is:

• A. \( \frac{3}{2} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{8} \)
• E. 1

Hint:

The AM-GM inequality states that for any positive numbers, the arithmetic mean is always greater than or equal to the geometric mean.

Answer 1

The Correct Option is A

The given inequality is of the form of the Arithmetic Mean-Geometric Mean (AM-GM) inequality. 
By applying AM-GM inequality: \[ a + b + c \geq 3 \left[ (a + b)(b + c)(c + a) \right]^{1/3} \] Thus, the maximum value of \( K \) occurs when the equality holds, which happens when: \[ K = \frac{3}{2} \]