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:
Show Hint
- \( \frac{3}{2} \)
- \( \frac{1}{2} \)
- \( \frac{1}{4} \)
- \( \frac{1}{8} \)
- 1
The Correct Option is A
Solution and Explanation
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: \[ K = \frac{3}{2} \]
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