Question

If \( a>0, b>0, c>0 \) and \( a, b, c \) are distinct, then \( (a + b)(b + c)(c + a) \) is greater than:

• A. \( 2(a + b + c) \)
• B. \( 3(a + b + c) \)
• C. \( 6abc \)
• D. \( 8abc \)

Hint:

Use the AM-GM inequality to compare products of sums for positive distinct values. This is helpful in many problems involving inequalities.

Answer 1

The Correct Option is D

Given \( a>0, b>0, c>0 \) and that \( a, b, c \) are distinct, the objective is to determine which expression is smaller than \( (a + b)(b + c)(c + a) \).
Step 1: Apply the AM-GM inequality.
The Arithmetic Mean is greater than or equal to the Geometric Mean: \[ AM \geq GM. \] For the terms \( a + b, b + c, c + a \), this yields: \[ (a + b)(b + c)(c + a) \geq 8abc. \] Therefore, the correct expression is \( 8abc \), corresponding to option (D).