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

We are given \( a>0, b>0, c>0 \) and the condition that \( a, b, c \) are distinct. The goal is to find which of the following expressions \( (a + b)(b + c)(c + a) \) is greater than. 
Step 1: Use AM-GM inequality
We know that the Arithmetic Mean is greater than or equal to the Geometric Mean: \[ AM \geq GM. \] Applying this to the terms \( a + b, b + c, c + a \), we get: \[ (a + b)(b + c)(c + a) \geq 8abc. \] Thus, the correct answer is \( 8abc \), which matches option (D).